paper stringlengths 9 16 | proof stringlengths 0 131k |
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math/0007127 | By symmetry, it suffices to show MATH. We may assume MATH is MATH-separable. By combining REF with REF , we may choose a power MATH of MATH, such that each element of MATH is a sum of MATH-separable elements of MATH of smaller degree. Thus, we may assume MATH is MATH-separable (and our bound MATH may depend on MATH). Define MATH as in the statement of REF , and let MATH be the codimension of MATH. Because MATH and MATH is MATH-separable, we have MATH, so REF implies MATH is bounded. Similarly, letting MATH be the MATH-separable element of MATH, and MATH be the MATH-separable element of MATH, we know that MATH is bounded. Then, because MATH we conclude that MATH is bounded. REF asserts that MATH is also bounded. |
math/0007127 | Choose MATH as in REF . Now choose MATH. Because MATH we have MATH so, from the choice of MATH and MATH, we conclude that MATH. |
math/0007127 | Because MATH the left-hand inequality is obvious. Let MATH and let MATH be the codimension of MATH. Then REF implies that there exist nonzero MATH with MATH, such that MATH contains a codimension-MATH subspace of MATH for MATH. Then, letting MATH, we have MATH, and MATH contains a codimension-MATH subspace of the ideal MATH. Thus, it suffices to show that the codimension of MATH in MATH is bounded above by MATH. Let MATH be the irreducible factors of MATH, so we may write MATH, MATH, and MATH, where MATH. From the NAME Remainder Theorem, we have MATH, so we may calculate the codimension in each factor, and then add them up. Fix MATH. By interchanging MATH and MATH if necessary, we may assume that MATH. It suffices to show that MATH thus (because MATH), we need only show that MATH. To show this, let MATH be minimal, such that MATH. (Obviously, we have MATH; we wish to show MATH.) Suppose MATH. (This will lead to a contradiction.) We have MATH, so MATH . This contradicts the minimality of MATH. |
math/0007127 | REF shows that, by replacing MATH with some MATH (using REF ), we may assume MATH, for every MATH. The terms MATH and MATH in REF are significant only when MATH is small. On the other hand, MATH can never be small (and nonzero) if MATH for some large MATH. Thus, by replacing MATH with some MATH (using REF ), we may assume MATH for every MATH. In particular, MATH if and only if MATH. Let MATH be the codimension of MATH in MATH. Choose some MATH . Choose a power MATH of MATH, such that MATH. There is some nonzero MATH, with MATH, such that MATH . Because MATH, we know that MATH is MATH-separable, so, by applying REF to MATH, we may assume MATH . By composing MATH with a map of the form MATH, for some MATH (with MATH), we may assume MATH and MATH, so MATH. Let MATH. It suffices to show MATH for every MATH. Suppose MATH, and let MATH . Let MATH, for any monic MATH with MATH. (Note that the definition of MATH implies that MATH is independent of the choice of MATH.) Assume MATH. Let MATH be any irreducible element of MATH with MATH. We claim that MATH contains a (monic) element MATH, such that MATH and MATH. To see this, let MATH with MATH. There is some MATH, such that MATH and MATH. Because MATH, this implies MATH, so MATH. Because MATH (and MATH), we know MATH. Because MATH is irreducible, we conclude that MATH. We also have MATH, so this implies MATH. Thus, we see that MATH is divisible by every irreducible polynomial over MATH of degree MATH, so MATH is divisible by MATH. Therefore MATH. However, we also know MATH (and all nonzero polynomials in MATH are monic, so MATH if MATH). This is a contradiction. Assume MATH. Choose some monic MATH, with MATH. By subtracting a polynomial of degree MATH, we may assume MATH; let MATH. There is some nonzero MATH with MATH, such that MATH. (Note that MATH.) Let MATH and MATH so MATH. Now, for each MATH, let MATH . For MATH, we have MATH, so MATH and MATH. Also, because MATH, we have MATH. Then, since MATH, we have MATH, so MATH . Also, for MATH, we have MATH so we see that MATH whenever MATH. Thus, we conclude that MATH . This is a contradiction. |
math/0007127 | Choose MATH as in REF . By replacing MATH with MATH and replacing MATH with MATH, we may assume MATH. Then, by composing MATH and MATH with MATH, we may assume MATH and MATH. Thus, MATH . We wish to show that there is some MATH, such that, for every MATH, we have MATH. For each MATH, there is some MATH, such that MATH. Fix MATH. Choose MATH as in REF , let MATH be the codimension of MATH, and choose MATH, such that MATH . Let MATH and choose some nonzero MATH, such that MATH and MATH. We have MATH . Thus, from the definition of MATH, we conclude that there is some MATH, such that MATH . Therefore MATH, so MATH. There is some MATH, such that MATH for every MATH. For MATH, let MATH denote the leading coefficient of MATH. Choose MATH, such that MATH generates MATH, that is, MATH. From REF , we know there is some MATH, such that MATH. We show MATH for every MATH. Given MATH, choose some MATH, such that MATH generates MATH, and such that MATH. From REF , there exist MATH, such that MATH and MATH. Because MATH, we have MATH and MATH. Thus, we have MATH . Because MATH generates MATH, we conclude that MATH. Therefore MATH . Similarly, we have MATH. Because we also have MATH, and MATH generates MATH, we conclude that MATH. Therefore MATH, as desired. |
math/0007127 | Let MATH, MATH be finite-index subgroups of MATH, such that MATH is an isomorphism. Let MATH, MATH be the image of MATH in MATH under the projection MATH with kernel MATH. By passing to a finite-index subgroup, we can assume that MATH. Since MATH, we can identify MATH with MATH, so MATH induces an isomorphism MATH. We can assume MATH for all MATH and MATH, such that MATH. For each nonzero MATH, let MATH. Note that MATH is a finite-index subgroup of MATH. For MATH, we have MATH if and only if MATH, so MATH. Thus, we can define a function MATH by MATH. Let MATH be such that MATH, and let MATH. Then MATH . Thus MATH . For any nonzero MATH and any MATH, since MATH is of finite index in MATH, we can find MATH so that MATH, MATH, and MATH. Then it follows from REF that MATH. Since MATH was arbitrary, we conclude that MATH . For an arbitrary MATH we can always find MATH so that MATH, thus we can define a function MATH, by MATH. REF implies that MATH is well defined. Note that MATH. Since MATH we have MATH. Since MATH is also an additive homomorphism, and MATH is an isomorphism, we conclude that MATH is a ring automorphism of MATH. Therefore MATH for MATH, where MATH, MATH, and MATH. Hence, by composing with the standard automorphism MATH, we obtain the claim. We may assume that MATH is the identity map. Let MATH with MATH. There is a finite-index subgroup MATH of MATH, such that MATH, for every MATH and MATH. Then, for all MATH, REF implies that MATH . Thus, choosing MATH, such that MATH, we have MATH . We conclude that MATH is constant, for MATH. By composing with a standard automorphism MATH, such that MATH, we may assume that MATH, so MATH. Then, by replacing MATH with a finite-index subgroup MATH, such that MATH, we may assume MATH. MATH can be extended to a conformally symplectic map MATH, with MATH. By REF , MATH for all MATH and MATH such that MATH. Because MATH is commensurable with MATH, this implies that MATH extends (uniquely) to a MATH-linear map MATH. For any MATH, we have MATH by REF . Because MATH spans MATH, this implies that MATH is conformally symplectic, with MATH. Completion of the proof. Define MATH by MATH. From REF , we see that MATH is an automorphism. Denote by MATH the map defined by MATH. Then MATH is a homomorphism and MATH, for MATH. |
math/0007127 | From REF , we may assume there exist CASE: a standard automorphism MATH of MATH; and CASE: a homomorphism MATH, such that MATH for all MATH. By REF , there exists a finite-index open subgroup MATH of MATH, containing MATH, such that MATH extends to MATH. Let MATH. Define MATH by MATH, so that MATH is a continuous homomorphism virtually extending MATH. Because MATH is trivial on MATH, we have MATH, so MATH is an automorphism. Because MATH, we see that MATH induces an isomorphism MATH. So MATH is an isomorphism. |
math/0007127 | Let MATH be an isomorphism, where MATH and MATH are arithmetic lattices in MATH. Define MATH and MATH . Then MATH. By passing to a finite-index subgroup we may assume that MATH, where MATH and MATH. Let MATH and MATH. Then, by passing to a finite-index subgroup, we may assume MATH and MATH. Let MATH denote the projection with kernel MATH. Then MATH virtually extends to a virtual automorphism MATH of MATH. It is easy to see that MATH is closed in MATH and hence is a lattice. Because MATH has finite index in MATH, we know MATH has finite index in MATH. Then, since MATH has finite index in MATH and MATH we conclude that MATH has finite index in MATH. Hence MATH is a lattice in MATH. By REF MATH virtually extends to a virtual automorphism MATH of MATH. Let MATH be the projection with kernel MATH, and let MATH. Then MATH virtual extends to a virtual automorphism of MATH. We claim that MATH is an arithmetic lattice in MATH. Because MATH and MATH, we have MATH . Then, because MATH, we conclude that MATH is a lattice in MATH. So the image of MATH in MATH is a lattice. Also, MATH so MATH is a lattice in MATH. Thus, we conclude that MATH is a lattice in MATH. Because MATH is contained in the arithmetic lattice MATH, this implies that MATH is arithmetic. From the preceding paragraph, we know that MATH is an isomorphism of arithmetic lattices in MATH. Let MATH denote the group isomorphism induced by the NAME automorphism MATH of the ground field MATH. Then there exist arithmetic lattices MATH in MATH, such that MATH and MATH, and an isomorphism MATH. By REF , we can virtually extend MATH to a virtual automorphism MATH of MATH. Then MATH is a virtual automorphism of MATH virtually extending MATH. Let MATH, so MATH is a virtual automorphism of MATH. We can define a map MATH on some finite index subgroup of MATH by MATH. By REF , MATH virtually extends to MATH. Then MATH is a virtual endomorphism of MATH. Since MATH we conclude (much as in the proof of REF ) that MATH is a virtual automorphism. It is easy to see that it virtually extends MATH. |
math/0007128 | Suppose that MATH has a nontrivial stabilizer MATH. Obviously MATH if and only if MATH, so suppose that MATH from now on. Since MATH is closed in MATH, it is compact and has a finite number of components. Suppose first that MATH is not discrete. Then the identity component of MATH must be a closed MATH-dimensional subgroup and hence conjugate to the subgroup MATH consisting of the rotations about the MATH-axis. Thus, MATH lies on the orbit of a cubic polynomial that is invariant under this rotation group. By replacing MATH by such an element, it can be supposed that the identity component of MATH is MATH. Consider the following four subspaces of MATH: Let MATH be the MATH-dimensional space spanned by MATH; let MATH be the MATH-dimensional space spanned by MATH and MATH; let MATH be the MATH-dimensional space spanned by MATH and MATH; and let MATH be the MATH-dimensional space spanned by MATH and MATH. Each of these subspaces is preserved by the elements of MATH. Moreover, MATH acts trivially on MATH, while the element MATH that represents rotation by an angle MATH about the MATH-axis, acts as rotation by the angle MATH on the MATH-dimensional space MATH for MATH. Obviously, the only nonzero elements of MATH that are fixed by MATH are those of the form MATH for some nonzero MATH. Moreover, since MATH is the unique linear factor of this polynomial, it follows that MATH, the stabilizer of this polynomial, must preserve the MATH-axis. Also, since this polynomial is positive on exactly one of the two rays in the MATH-axis emanating from the origin, it follows that MATH must also fix the orientation of the MATH-axis. Thus, MATH. Moreover, note that by a rotation that reverses the MATH-axis, the element MATH is carried into the element MATH. Thus, one can assume that MATH. Now suppose that MATH is discrete (and hence finite). Let MATH be an element of finite order MATH. Then MATH is rotation about a line by an angle of the form MATH for some integer MATH relatively prime to MATH and satisfying MATH. Replacing MATH by an element in its MATH-orbit, I can assume that the fixed line of MATH is the MATH-axis. Since the action of MATH on MATH is a rotation by the angle MATH for MATH, it follows that, unless either MATH or MATH are integers, then the only elements of MATH that are fixed by MATH are the elements of MATH. Since these elements have a continuous symmetry group, and so, by hypothesis, cannot be MATH, it follows that either MATH or MATH are integers, that is, that MATH or MATH. If MATH, then MATH must lie in MATH, that is, there must be constants MATH, MATH, and MATH, so that MATH . By a rotation that reverses the MATH-axis, if necessary, I can assume that MATH and then, by applying a rotation in MATH, I can assume that MATH and MATH. Since MATH is discrete, MATH cannot be zero, so MATH. Note that MATH is a rotation by an angle of MATH about the MATH axis, and that this certainly preserves any MATH in the above form. Note also that every such MATH has a linear factor. In particular, to each element MATH of order MATH in MATH, there corresponds a linear factor of MATH that is fixed (up to a sign) by MATH. If MATH, then MATH, and it is clear that the elements of MATH must permute the planes MATH, MATH, and MATH. It follows that MATH must be the group MATH of order MATH described in the proposition. If MATH, then it is still true that MATH has a linear factor, that is, MATH . When MATH, the quadratic factor in the above expression is irreducible (since MATH and MATH are positive), so MATH must stabilize the MATH-axis. In fact, since MATH is positive on the positive ray of the MATH-axis, MATH must actually be a subgroup of MATH. Since MATH, MATH must therefore be isomorphic to MATH, generated by the rotation by MATH about the MATH-axis. On the other hand, when MATH, the polynomial MATH factors as MATH . These three linear factors of MATH are linearly dependent, so that MATH vanishes on the union of three coaxial planes that meet pairwise at an angle of MATH. Consequently, MATH lies on the MATH-orbit of an element of the form MATH where MATH. Since MATH must preserve these factors up to a sign, MATH is isomorphic to MATH and is generated as claimed in the proposition. Finally, assume that MATH has no element of order MATH. Then, by the above argument, all of the nontrivial elements of MATH have order MATH. By the well-known classification of the finite subgroups of MATH, it follows that MATH must be isomorphic to MATH. Let MATH be a generator of MATH and assume (as one may, by replacing MATH by an element in its MATH-orbit) that MATH is rotation by an angle of MATH about the MATH-axis. Then the elements of MATH that are fixed by MATH are the elements in MATH, that is, those of the form MATH . By a rotation about the MATH-axis, MATH can be replaced by an element in its orbit that is of the above form but that satisfies MATH and MATH. Now, MATH, since, otherwise the stabilizer of MATH would contain MATH. After rotation by an angle of MATH about the MATH-axis if necessary, I can further assume that MATH. In fact, MATH, since, otherwise, MATH would be isomorphic to MATH, contrary to hypothesis. It remains to determine those positive values of MATH and MATH (if any) for which MATH has a symmetry group larger than MATH. If the symmetry group MATH is to be larger than MATH, then, by the aforementioned classification, either MATH contains an element of order MATH or MATH is infinite. In either case, by the above arguments, MATH must have a linear factor. Now, it is straightforward to verify that MATH has no linear factor unless MATH. Thus, the stabilizer is MATH except in this case. On the other hand MATH and the three linear factors vanish on three mutually orthogonal MATH-planes. It follows immediately that this MATH lies on the orbit of MATH for some MATH. |
math/0007128 | By REF , any cubic that has a symmetry of order MATH has a linear factor. Conversely, suppose that MATH has a linear factor and is nonzero. By applying a MATH symmetry, it can be assumed that MATH divides MATH, implying that MATH has the form MATH which clearly has a symmetry of order MATH that fixes MATH. The quadratic factor is reducible if and only if either MATH, in which REF rotation in the MATH-plane reduces MATH to zero, so that the symmetry group is MATH, or else MATH, in which case MATH factors into three linearly dependent factors, so that the symmetry group is MATH. |
math/0007128 | A cubic MATH is linear in a direction MATH if and only if the direction generated by MATH is a singular point of the projectivized curve MATH in MATH. Thus, the first statement follows, since the set of real cubic curves with a real singular point is a semi-analytic set of codimension MATH. If there are two distinct singular points, then the curve MATH must be a union of a line with a conic. If there are three distinct singular points, then the curve MATH must be the union of three nonconcurrent lines. Further details are left to the reader. |
math/0007128 | Let MATH satisfy the hypotheses of the theorem. If the fundamental cubic MATH vanishes identically, then MATH is a MATH-plane, so assume that it does not. The locus where MATH vanishes is a proper real-analytic subset of MATH, so its complement MATH is open and dense in MATH. Replace MATH by a component of MATH, so that it can be assumed that MATH is nowhere vanishing on MATH. By REF , since the stabilizer of MATH is MATH for all MATH, there is a positive (real-analytic) function MATH with the property that the equation MATH defines a MATH-subbundle MATH of the adapted coframe bundle MATH. On the subbundle MATH, the following identities hold: MATH . Moreover, because MATH is a MATH-bundle, relations of the form MATH hold on MATH for some functions MATH. Moreover, for MATH, MATH, MATH there exist functions MATH on MATH so that MATH . Substituting the relations REF, and REF into the identities MATH and using the identities MATH then yields polynomial relations among these quantities that can be solved, leading to relations of the form MATH where, for brevity, I have written MATH for MATH. Note that REF implies that MATH. Differentiating the last equation in REF implies that there exists a function MATH on MATH so that MATH . Substituting REF into the identities MATH and expanding, again using the identities MATH, yields the relations MATH . Differentiating these last equations yields only identities. The structure equations found so far can be summarized as follows: MATH is a MATH bundle on which REF-forms MATH are a basis. They satisfy the structure equations MATH and the exterior derivatives of these equations are identities. These equations imply that MATH. Since MATH and MATH are connected, it follows that there is a constant MATH so that MATH. Consequently, there is a function MATH that is well-defined on MATH that satisfies MATH and the bound MATH. It then follows from the last two equations of REF that MATH . Moreover, setting MATH for MATH and MATH yields MATH which are the structure equations of the metric of constant curvature MATH on MATH. Conversely, if MATH is the metric of constant curvature MATH on MATH, then, on the product MATH, consider the quadratic and cubic forms defined by MATH . The metric MATH is complete and the pair MATH satisfy the NAME and NAME equations that ensure that MATH can be isometrically embedded as a special Lagrangian MATH-fold in MATH inducing MATH as the fundamental cubic. Thus, for each value of MATH, there exists a corresponding special Lagrangian MATH-fold that is complete and unique up to special Lagrangian isometries of MATH. Since the parameter MATH is accounted for by dilation in MATH, it now follows that these special Lagrangian MATH-folds are the NAME examples, as desired. Note that since these are complete and since MATH is nowhere vanishing, it follows that MATH for the NAME examples, and hence for all examples. |
math/0007128 | Let MATH be a connected special Lagrangian submanifold with the property that its fundamental cubic MATH has a MATH-symmetry at each point. If MATH vanishes identically, then MATH is an open subset of a special Lagrangian MATH-plane, so assume that it does not. Let MATH be the dense open subset where MATH is nonzero. By REF , since the stabilizer of MATH is MATH for all MATH, there is a positive (real-analytic) function MATH for which the equation MATH defines a MATH-subbundle MATH over MATH of the adapted coframe bundle MATH. On MATH, the following identities hold: MATH . Since MATH is a MATH-bundle over MATH, there are relations MATH holding on MATH for some functions MATH. Moreover, there exist functions MATH for MATH on MATH so that MATH . Substituting the relations REF, and REF into the identities MATH and using the identities MATH then yields polynomial relations among these quantities that can be solved, leading to relations MATH . Substituting REF into the identities MATH yields MATH, contrary to hypothesis. |
math/0007128 | Suppose that MATH satisfies the hypotheses of the theorem. If the fundamental cubic MATH vanishes identically on MATH, then MATH is a MATH-plane and there is nothing to show, so suppose that MATH. Let MATH be the open dense subset where MATH. The hypothesis that MATH has MATH-symmetry at every MATH implies that there is a positive function MATH and a MATH-subbundle MATH over MATH with projection MATH on which the identity MATH holds. In particular, the second fundamental form of MATH has the form MATH where MATH are the vector-valued functions defined by the moving frame relation MATH. It follows from REF that MATH is an austere submanifold of dimension MATH. By REF, it follows that either MATH is locally the product of a line in MATH with a minimal surface MATH in the orthogonal MATH-plane, or else there exists a dense open subset MATH so that every point of MATH has an open neighborhood in MATH that lies in a twisted cone constructed as in REF from a minimal immersion MATH and an auxiliary function MATH satisfying REF. Since the group of translations and MATH-rotations in MATH acts transitively on the space of lines, it follows that if MATH is locally an orthogonal product MATH and is special Lagrangian, then, up to translation by a constant, MATH must be a complex curve in the complex MATH-plane MATH orthogonal to the linear factor, where the complex structure on MATH is taken to be as defined in REF . On the other hand, if MATH is not locally an orthogonal product and so is a twisted cone as described above, then one sees from the formula for MATH derived in REF that the immersion MATH must not only be minimal, but must have the property that MATH as well, as desired. |
math/0007128 | It suffices to assume that MATH is connected, so do this. If any MATH-leaf MATH is planar, even locally, then this plane must be MATH-isotropic and NAME and NAME 's REF implies that MATH itself must contain an open subset of a special Lagrangian MATH-plane. By real-analyticity, it follows that MATH itself is planar and hence that MATH vanishes identically for all MATH. Thus, from now on, I can assume that none of the MATH-leaves are planar and that MATH itself is nonplanar. Choose MATH and restrict MATH to a neighborhood MATH on which the foliation can be expressed a product, that is, MATH for some open domain MATH, and the MATH-leaves in MATH are of the form MATH for MATH. Then, by hypothesis, for each MATH, there exists a unique real MATH-plane MATH so that MATH, and the surface MATH is MATH-isotropic. Since the surface MATH is non-planar, the plane MATH itself must Lagrangian, although it cannot be special Lagrangian, since, otherwise, the uniqueness aspect of NAME and NAME 's Theorem MATH would imply that MATH, contradicting the assumption that MATH is not planar. It is not difficult to see that the curve MATH must be smooth, since the foliation MATH is assumed to be smooth. Now, consider the MATH-subbundle MATH over MATH with the property that the vector-valued functions MATH and MATH are an oriented basis of the tangent space to the MATH-leaves. Then MATH is well-defined on MATH and vanishes when pulled back to any MATH-leaf. Now, the set of Lagrangian planes that contain MATH and MATH is the circle of MATH-planes that contain MATH, MATH and that are contained in the span of MATH. In particular, MATH lies in this plane for each leaf MATH. Since each leaf MATH lies in MATH, it follows that the second fundamental form MATH has the property that MATH and MATH must vanish when restricted to REF-planes defined by MATH, that is, it must be true that MATH and MATH are multiples of MATH. However, by NAME 's homogeneity relation MATH it now follows that MATH itself must be a multiple of MATH, that is, MATH is reducible at every point of MATH, as desired. Finally, by REF , the MATH-stabilizer of MATH must contain an element of order MATH for all MATH. |
math/0007128 | By assumption, at a generic point MATH, the MATH-stabilizer subgroup of MATH is isomorphic to MATH. Let MATH be the open, dense subset where this holds. Then by REF , there exist positive functions MATH with MATH and a MATH-subbundle MATH over MATH on which the following identity holds: MATH . (Of course, MATH is a double cover and the reader can just think of the coframing MATH as being well-defined on MATH up to the ambiguity of replacing MATH and MATH by MATH and MATH.) Consequently, on the subbundle MATH, the following identities hold: MATH . Moreover, because MATH is a MATH-bundle, relations of the form MATH hold on MATH for some functions MATH. Moreover, for MATH there exist functions MATH and MATH on MATH so that MATH . Substituting the relations REF, and REF into the identities MATH and using the identities MATH then yields polynomial relations among these quantities that can be solved, leading to relations of the form MATH where, for brevity, I have introduced the notation MATH . Using REF to expand out the identities MATH yields relations on the exterior derivatives of MATH, MATH, and MATH. These can be expressed by the condition that there exist functions MATH, MATH, and MATH so that the equations MATH hold. Substituting REF into the identities MATH and expanding, again using the identities MATH, yields MATH . Finally, expanding out the identities MATH shows that they are equivalent to the formula MATH . The exterior derivative of REF is an identity. For future use, I record the formulae MATH which follow from the identities MATH coupled with REF. At this point, it is worthwhile taking stock of what has been accomplished. Consider the system of quantities MATH . The formulae REF, and REF express the exterior derivatives of these quantities as polynomials in these quantities. Moreover, the relation MATH for MATH any one of these quantities follows by formal expansion and use of the given exterior derivative formulae. By REF, for any six constants MATH, there exists an open neighborhood MATH of MATH that is endowed with three linearly independent MATH-forms MATH and six functions MATH that satisfy REF , and REF and also satisfy MATH . Moreover these functions and forms are real-analytic and unique in a neighborhood of MATH, up to a real-analytic local diffeomorphism fixing MATH. Now, given such a system MATH on a simply connected MATH-manifold MATH, one can set MATH, define MATH by the last three equations of REF, define MATH by REF , and see that the affine structure equations MATH are identities. Thus, there is an immersion of MATH, unique up to translation and MATH-rotation, as a special Lagrangian MATH-manifold in MATH that induces these structure equations. In particular, it follows that the space of germs of special Lagrangian MATH-manifolds in MATH whose fundamental cubics are of the form REF is of dimension MATH. Moreover, any two that agree to order MATH at a single point must be equal in a neighborhood. It is not difficult to argue from this that the space one gets by reducing modulo the equivalence relation defined by analytic continuation is a MATH-dimensional singular space. Now, the first of REF shows that the MATH-plane field MATH is integrable, moreover, the structure equations found so far imply MATH . In particular, the MATH-plane MATH is constant along each leaf of MATH and, moreover each such leaf lies in an affine MATH-plane parallel to this MATH-plane. Thus, all of these examples are foliated in codimension MATH by MATH-plane sections. Moreover, an examination of the structure equations shows that the space of congruence classes of such MATH-plane sections is of dimension MATH, the same as the dimension of quadric surfaces in MATH-space. In fact, using the structure equations, it is not difficult to show that these MATH-plane sections are, in fact, quadric surfaces. For the sake of brevity, I will not include the details of this routine calculation here. It follows that these special Lagrangian MATH-folds all belong to the class of NAME examples, as extended by NAME. |
math/0007128 | By REF , any such MATH must have a reducible fundamental cubic MATH. Thus, the MATH-stabilizer of MATH at each point contains a MATH and so is either isomorphic to MATH, MATH, or MATH. If this stabilizer is isomorphic to MATH at a generic point, then REF applies, showing that MATH is a NAME example. If this stabilizer is isomorphic to MATH at a generic point, the discussion at the end of REF shows that the only such examples that are foliated in codimension MATH by MATH-plane sections have the property that these sections are necessarily (possibly singular) quadric surfaces, so that such a MATH is, again, a NAME example. Finally, if the stabilizer is isomorphic to MATH at a generic point, then REF applies. |
math/0007128 | Let MATH satisfy the hypotheses of the theorem. The locus of points MATH for which the MATH-stabilizer of MATH is larger than MATH is a proper real-analytic subset of MATH, so its complement MATH is open and dense in MATH. Thus, I can, without loss of generality, replace MATH by a component of MATH. In other words, I can assume that the MATH-stabilizer of MATH is isomorphic to MATH for all MATH. By REF , since the stabilizer of MATH is MATH for all MATH, there are positive (real-analytic) functions MATH and MATH on MATH with the property that the equation MATH defines a MATH-subbundle MATH of the adapted coframe bundle MATH. Moreover, the expression MATH is nowhere vanishing on MATH. Now, on the subbundle MATH, the following identities hold: MATH . Moreover, because MATH is a MATH-bundle, relations of the form MATH hold on MATH for some functions MATH. Moreover, for MATH there exist functions MATH and MATH on MATH so that MATH . Substituting the relations REF, and REF into the identities MATH and using the identities MATH then yields polynomial relations among these quantities that can be solved, leading to relations of the form MATH where, for brevity, I have introduced the notation MATH . Using REF to expand out the identities MATH and also the identities MATH yields relations on the exterior derivatives of MATH, MATH, and MATH. When these are solved, one finds that there are functions MATH and MATH so that the equations MATH hold where MATH . Observe that, if one sets MATH in the current structure equations, then these become, up to a trivial change of notation, the same structure equations as those for the special Lagrangian cones discussed in REF. This is a first hint that these examples must be related to the special Lagrangian cones. The next observation is that the structure equations MATH are identical (after replacing MATH by MATH) to the last two equations of REF. In particular, there must exist a constant MATH and a function MATH on MATH satisfying the bound MATH so that MATH . It then follows from REF that MATH . By dilation in MATH, one can reduce to the case MATH, so assume this from now on. Consider the following expressions: MATH . The structure equations derived above show that MATH where MATH. In particular, MATH. Let MATH be fixed and let MATH be a MATH-neighborhood on which there exist functions MATH and MATH vanishing at MATH that satisfy MATH and MATH. Then the functions MATH are independent on MATH and, by shrinking MATH if necessary, I can assume that MATH is a product open set of the form MATH where MATH is a connected interval and MATH is a disc centered on the origin. Of course, the functions MATH, MATH, MATH, MATH and MATH can be regarded as functions on MATH, since their differentials are linear combinations of MATH and MATH. In fact, these functions and forms can now be regarded as defined on the open set MATH by simply reading the formulae above backwards. Thus, for example MATH and so forth. This gives quantities MATH, MATH, MATH, MATH, and MATH that are well-defined on all of MATH and that satisfy the originally derived structure equations. It follows that there is an immersion of MATH into MATH as a special Lagrangian MATH-fold that extends MATH and pulls back the constructed forms and quantities to agree with the given ones on MATH. The chief difference is that each of the MATH-curves in MATH is mapped to a complete curve in MATH. Next, observe that the equations MATH which are identical to the corresponding equations in REF, then show that the leaves of the curve foliation defined by MATH are congruent to the leaves of the corresponding foliation by the MATH-curves in REF. Finally, note that, setting MATH that is, MATH and MATH in the above structure equations on MATH gives an immersion of MATH into MATH with the property that the cone on the image MATH is a special Lagrangian MATH-fold. Because the MATH-curves meet this surface orthogonally, it follows easily that the image of MATH is exactly MATH as described in REF . Further details are left to the reader. |
math/0007128 | This is a straightforward application of the NAME Theorem CITE so I will only give the barest details. This is a local result, so it suffices to give a local proof. Let MATH be the rank of MATH and let MATH be its codimension. For any point MATH, there is an open MATH-neighborhood MATH on which there exist real-analytic MATH-forms MATH with real values and MATH with complex values with the property that the equations MATH define the restriction of MATH to MATH and with the property that MATH are complex linear on MATH and are linearly independent over MATH at each point of MATH. There are identities of the form MATH . REF is simply that the functions MATH all vanish identically. Under this hypothesis, the real-analytic exterior differential system MATH generated algebraically by the MATH and the real and imaginary parts of the MATH-forms MATH is involutive and each of the MATH-dimensional integral elements is regular and lies in a unique MATH-dimensional integral element. Now apply the NAME theorem. |
math/0007128 | This is immediate from the formulae for MATH and MATH. |
math/0007128 | I will only sketch the proof, since the details are straightforward. First, the easy direction: If MATH is ruled, then the analytic set MATH must have dimension MATH at least. Conversely, if the dimension of MATH is at least MATH, then it contains an immersed analytic arc MATH, which generates a ruled surface MATH for some appropriate domain MATH. The surface MATH must be MATH-isotropic since MATH is Lagrangian. Thus, the arc MATH must be a MATH-curve. By REF , this arc lies in a MATH-holomorphic surface MATH. By REF , there is a dense open region MATH so that MATH is an immersed ruled special Lagrangian MATH-fold. It is not hard to see that this MATH contains at least an open subset of MATH. Since by NAME and NAME 's REF , the real-analytic MATH-isotropic surface MATH lies in a locally unique special Lagrangian MATH-fold, it follows that MATH and MATH must intersect in an open set. Thus MATH is ruled on an open set. By real-analyticity and connectedness, it must be ruled everywhere. |
math/0007128 | Combine REF . |
math/0007128 | First, I will define the almost NAME on MATH and show that it is NAME. Consider the mapping MATH that sends the coframe MATH to the oriented line spanned by MATH that passes through MATH. Since the structure equations give MATH it follows that the ten MATH-forms that appear on the right-hand side of this equation are MATH-semibasic and it is evident that MATH while MATH. The fibers of MATH are cosets of the subgroup of the motion group that fixes an oriented line in MATH and hence are diffeomorphic to MATH. In particular, they are connected. Define complex-valued MATH-forms on MATH by MATH . These forms are MATH-semibasic and satisfy the equations MATH while MATH . Since the fibers of MATH are connected, it follows that there is a (unique) complex structure MATH so that the complex-valued MATH-forms on MATH that are MATH-linear on MATH pull back to be linear combinations of MATH. Moreover, REF imply that the almost NAME MATH is NAME, as promised. This structure is clearly real-analytic since it is homogeneous under the action of the complex isometry group on MATH. (Note also, by the way, that REF also imply that this almost NAME is not integrable.) This completes the proof of REF. Now suppose that MATH is a ruled special Lagrangian MATH-fold that is not a MATH-plane. Then, on a dense open set, this ruling can be chosen to be real-analytic and smooth. Consider the subbundle MATH of the adapted frame bundle over MATH that has MATH tangent to the ruling direction. Thus, the curves in MATH defined by the differential equations MATH are straight lines and, of course, MATH is tangent to these straight lines. It follows that MATH. (In fact, this is necessary and sufficient that the MATH-integral curves be straight lines in MATH.) Since MATH it follows, in particular, that MATH. Since MATH, it follows from this that MATH for MATH, MATH, and MATH. In particular, the fundamental cubic MATH is linear in the direction MATH. Of course, by REF , it follows that, at points where MATH is non-zero, it is linear in at most three directions. Moreover, by REF , there is no non-planar special Lagrangian MATH-fold whose cubic is linear in three directions. Thus, either MATH is linear in exactly two directions on a dense open set, or else it is linear in exactly one direction on a dense open set. If MATH is linear in exactly two directions on a dense open set, then, again by REF , it follows that MATH is reducible at every point and, on a dense open set, cannot have a MATH-stabilizer isomorphic to MATH, since these are not linear in two distinct variables. It follows that the MATH-stabilizer at a generic point is MATH, so that, by REF , MATH must be one of the NAME examples. Moreover, the two linearizing directions, since they represent singular points of the projectivized cubic curve, must lie on the linear factor of MATH. Thus, the two possible rulings must lie in the MATH-dimensional slices by MATH-planes. Of course, this can only happen if the quadrics that are these slices are doubly ruled. Conversely, if the quadrics that are these slices are doubly ruled, then, obviously, MATH must be doubly ruled as well. This establishes Item MATH (as well as the fact that a non-planar special Lagrangian MATH-fold cannot be triply ruled). At any rate, note that MATH and that MATH where I have used the symmetry and trace conditions on MATH together with the condition MATH. It follows that MATH . There are now two cases to deal with. Either MATH and MATH vanish identically or they do not. Suppose first that MATH. In this case, one can, after restricting to a dense open set, adapt frames so that the fundamental cubic has the form MATH where MATH. In particular, the MATH-stabilizer of MATH at the generic point is MATH. Set MATH, so that the notation agrees with the notation established in REF. Looking back at the structure equations from that section, one sees that MATH . In particular, MATH. Since it has already been established that, in this case, MATH it follows immediately that the natural map from the frame bundle to MATH that sends a coframe MATH to MATH maps the coframe bundle into a MATH-holomorphic surface and that this surface is simply the space of lines of the ruling. Now suppose that MATH and MATH do not vanish identically. Then, by restricting to the dense open set where they are not simultaneously zero, we can reduce frames to arrange that MATH, but that MATH. In fact, there will exist functions MATH, MATH, and MATH so that MATH . This reduces the frames to a finite ambiguity, but I will not worry about this, since it does not impose any essential difficulty. Of course, MATH and MATH cannot vanish identically by REF . In particular, on this adapted bundle, the following formulae hold: MATH . Now, there are functions MATH and MATH, MATH, and MATH so that MATH . Just as in previous cases of moving frame analyses, substituting these equations into the structure equations for MATH yields REF equations on these REF quantities. I will not give the whole solution, since that is not needed for this argument, but will merely note that these equations imply MATH and that MATH while MATH. In particular, this implies MATH just as in the first case. Moreover, since MATH and MATH, it also follows that MATH . Since it has already been shown that MATH it follows, once again, that the natural map from the frame bundle to MATH that sends a coframe MATH to MATH maps the coframe bundle into a MATH-holomorphic surface and that this surface is simply the space of lines of the ruling. Thus, it has been shown that any ruled special Lagrangian MATH-fold is locally the MATH-fold generated by a MATH-holomorphic surface in MATH. The only thing left to check is that every MATH-holomorphic surface in MATH generates a special Lagrangian MATH-fold in MATH. However, given the analysis already done, this is an elementary exercise in the moving frame and can be safely left to the reader. |
math/0007130 | The strategy of proof is the same as in REF-dimensional case. One starts with an arbitrary sequence of asymptotically holomorphic sections of MATH over MATH, and perturbs it first to obtain the transversality properties. Provided that MATH is large enough, each transversality property can be obtained over a ball by a small localized perturbation, using the local transversality result of NAME REF . A globalization argument then makes it possible to combine these local perturbations into a global perturbation that ensures transversality everywhere REF . Since transversality properties are open, successive perturbations can be used to obtain all the required properties : once a transversality property is obtained, subsequent perturbations only affect it by at most decreasing the transversality estimate. CASE: One first obtains the transversality statements in REF, MATH and MATH of REF ; as in REF-dimensional case, these properties are obtained for example, simply by applying the main result of CITE. Observe that all required properties now hold near the base locus MATH of MATH, so we can assume in the rest of the argument that the points of MATH being considered lie away from MATH, and therefore that MATH is locally well-defined. One next ensures condition MATH, for which the argument is an immediate adaptation of that in REF, the only difference being the larger number of coordinate functions. CASE: The next property we want to get is REF. Here a significant generalization of the argument in REF is needed. The problem reduces, as usual, to showing that the uniform transversality to MATH of MATH can be ensured over a small ball centered at a given point MATH by a suitable localized perturbation. As in CITE one can assume that MATH is of the form MATH and therefore locally trivialize MATH via the quasi-isometric map MATH ; this reduces the problem to the study of a MATH-valued map MATH. Because MATH is bounded from below, we can assume (after a suitable rotation) that MATH is greater than some fixed constant. Also, fixing suitable approximately holomorphic NAME coordinates MATH (using REF, which trivially extends to dimensions larger than REF), we can after a rotation assume that MATH is of the form MATH, where the complex number MATH is bounded from below. By REF, there exist asymptotically holomorphic sections MATH of MATH with exponential decay away from MATH. Define the asymptotically holomorphic REF-forms MATH for MATH. At MATH, REF-form MATH is proportional to MATH ; therefore, over a small neighborhood of MATH, the transversality to MATH of MATH in the sense explained above is equivalent to the transversality to MATH of the projection of MATH onto the subspace generated by MATH. In terms of MATH-jets, REF-forms MATH define a local frame in the normal bundle to the stratum of non-regular maps at MATH. Now, express MATH in the form MATH over a neighborhood of MATH, where MATH are complex-valued functions and MATH has no component along MATH. Then, the transversality to MATH of MATH is equivalent to that of the MATH-valued function MATH. Since the functions MATH are asymptotically holomorphic, using suitable NAME coordinates at MATH we can use REF to obtain, for large enough MATH, the existence of constants MATH smaller than any given bound MATH and such that MATH is MATH-transverse to MATH over a small ball centered at MATH, where MATH (MATH is a fixed constant). Letting MATH and calling MATH and MATH the projective map defined by MATH and the corresponding local MATH-valued map, we get that MATH, and therefore that MATH is transverse to MATH near MATH. Since the perturbation of MATH has exponential decay away from MATH, we can apply the standard globalization argument to obtain property MATH everywhere. CASE: The next properties that we want to get are MATH and MATH. It is possible to extend the arguments of CITE and CITE to the higher dimensional case ; however this yields a very technical and lengthy argument, so we outline here a more efficient strategy following the ideas of CITE. Thanks to the previously obtained transversality properties MATH and MATH, both MATH and MATH are well-defined over a neighborhood of MATH, so the statements of MATH and MATH are well-defined. Moreover, observe that REF implies property MATH, because at any point where MATH vanishes, MATH necessarily vanishes as well, and if it does so transversely then the same is true for MATH as well. So we only focus on MATH. This property can be rephrased in terms of transversality to the codimension MATH stratum MATH in the bundle MATH of holomorphic REF-jets of maps from MATH to MATH. However this stratum is singular, even away from the substratum MATH corresponding to the non-transverse vanishing of MATH ; in fact it is reducible and comes as a union MATH, where MATH is the stratum corresponding to non-immersed points of the branch curve, and MATH is the stratum corresponding to tangency points of the branch curve. Therefore, one first needs to ensure transversality with respect to MATH, which is a smooth codimension MATH stratum (``vertical cusp points of the branch curve") away from MATH. CASE: We first show that a small perturbation can be used to make sure that the quantity MATH remains bounded from below, that is, that given any point MATH, either MATH is larger than a fixed constant, or MATH lies at more than a fixed distance from MATH, or MATH lies close to a point of MATH where MATH is larger than a fixed constant. Since this transversality property is local and open, we can obtain it by successive small localized perturbations, as for the previous properties. Fix a point MATH, and assume that MATH is small (otherwise no perturbation is needed). By REF, we know that necessarily MATH is bounded away from zero at MATH ; a rotation in the first two coordinates makes it possible to assume that MATH and MATH is bounded from below near MATH. As above, we replace MATH by the MATH-valued map MATH, where MATH. By assumption, we get that MATH is small. This implies in particular that MATH is small at MATH, and therefore property MATH gives a lower bound on its covariant derivative. Moreover, by REF we also have a lower bound on MATH, which after a suitable rotation can be assumed equal to MATH for some MATH. So, as above we can express MATH by looking at its components along MATH for MATH ; we again define the MATH-forms MATH, and the functions MATH are defined as previously. Define a MATH-form MATH over a neighborhood of MATH by MATH : at points of MATH, the vanishing of MATH is equivalent to that of MATH, or equivalently to that of MATH. So our aim is to show that the quantity MATH, which is a section of a rank MATH bundle MATH near MATH, can be made bounded from below by a small perturbation. For this purpose, we first show the existence of complex-valued polynomials MATH and local sections MATH of MATH, MATH, such that : REF for any coefficients MATH, replacing the given sections of MATH by MATH affects MATH by the addition of MATH ; REF the sections MATH define a local frame in MATH, and MATH is bounded from below by a universal constant. First observe that, by REF, MATH is bounded from below near MATH, whereas we may assume that MATH is small (otherwise no perturbation is needed). Therefore, MATH (which at MATH is colinear to MATH) lies close to the span of the MATH. In particular, after a suitable rotation in the MATH last coordinates on MATH, we can assume that MATH is small at MATH. On the other hand, we know that there exists MATH such that MATH lies far from the span of the MATH. We then define MATH and MATH. Adding to MATH a quantity of the form MATH does not affect MATH, but affects MATH by the addition of a non-trivial multiple of MATH, and similarly affects MATH by the addition of a non-trivial multiple of MATH. The other MATH are not affected. Therefore, MATH changes by an amount of MATH where the constants MATH and MATH are bounded from above and below. The first term is bounded from below by construction, while the second term is only present if MATH (this requires MATH), and in that case it is small because MATH is small. Therefore, the local section MATH of MATH naturally corresponding to such a perturbation is of the form MATH at MATH, where MATH is bounded from below. Next, for MATH we define MATH and MATH, and observe that adding MATH to MATH affects MATH by adding a nontrivial multiple of MATH. Therefore, the local section of MATH corresponding to this perturbation is at MATH of the form MATH, where MATH is a constant bounded from below. It follows from this argument that the chosen perturbations MATH and MATH for MATH, and the corresponding local sections MATH of MATH, satisfy REF expressed above. Observe that, because MATH define a local frame at MATH and MATH is bounded from below at MATH, the same properties remain true over a ball of fixed radius around MATH. Now that a local approximately holomorphic frame in MATH is given, we can write MATH in the form MATH for some complex-valued functions MATH ; it is easy to check that these functions are asymptotically holomorphic. Therefore, we can again use REF to obtain, if MATH is large enough, the existence of constants MATH smaller than any given bound MATH and such that MATH is bounded from below by MATH (MATH is a fixed constant) over a small ball centered at MATH. Letting MATH and calling MATH, MATH and MATH the projective map defined by MATH and the corresponding local maps, we get that MATH is by construction bounded from below by MATH, for a fixed constant MATH ; indeed, observe that the non-linear term MATH in the perturbation formula does not play any significant role, as it is at most of the order of MATH. Since the perturbation of MATH has exponential decay away from MATH, we can apply the standard globalization argument to obtain uniform transversality to the stratum MATH everywhere. CASE: We now obtain uniform transversality to the stratum MATH. The strategy and notations are the same as above. We again fix a point MATH, and assume that MATH lies close to a point of MATH where MATH is small (otherwise, no perturbation is needed). As above, we can assume that MATH is bounded from below and define a MATH-valued map MATH. Two cases can occur : either MATH is bounded away from zero, or it is small and in that case by REF we know that MATH is bounded from below near MATH. We start with the case where MATH is bounded from below; in other words, we are not dealing with tangency points but only with cusps. In that case, we can use an argument similar to REF , except that the roles of the two components of MATH are reversed. Namely, after a rotation we assume that MATH for some nonzero constant MATH, and we define components MATH of MATH as previously (using MATH rather than MATH to define the MATH). Let MATH : along MATH, the ratio between MATH and MATH, or equivalently MATH, is bounded between two fixed constants, so the transverse vanishing of MATH is what we are trying to obtain. More precisely, our aim is to show that the quantity MATH, which is a section of a rank MATH bundle MATH near MATH, can be made uniformly transverse to MATH by a small perturbation. For this purpose, we first show the existence of complex-valued polynomials MATH and local sections MATH of MATH, MATH, such that : REF for any coefficients MATH, replacing the given sections of MATH by MATH affects MATH by the addition of MATH ; REF the sections MATH define a local frame in MATH, and MATH is bounded from below by a universal constant. By the same argument as in REF , we find after a suitable rotation an index MATH such that, letting MATH and MATH, the corresponding local section MATH of MATH is, at MATH, of the form MATH, with MATH bounded from below by a fixed constant. Moreover, adding MATH to MATH amounts to adding MATH to MATH and does not affect the other MATH's, by the argument in REF . So, letting MATH and MATH, we get that the corresponding local sections of MATH are of the form MATH, where the coefficient MATH is in MATH-th position. So it is easy to check that both REF are satisfied by these perturbations. The rest of the argument is as in REF : expressing MATH as a linear combination of MATH, one uses REF to obtain transversality to MATH over a small ball centered at MATH. We now consider the second possibility, namely the case where MATH is small, which corresponds to tangency points. By REF we know that MATH is bounded from below, and we can assume that it is colinear to MATH. We then define components MATH of MATH as usual (as in REF and unlike the previous case, the MATH are defined using MATH rather than MATH). Letting MATH, we want as before to obtain the transversality to MATH of the quantity MATH, which is a local section of a rank MATH bundle MATH near MATH. For this purpose, as usual we look for polynomials MATH, MATH and local sections MATH satisfying the same REF as above. In order to construct MATH, observe that, by the result of REF , the quantity MATH is bounded from below at MATH. So, adding to MATH a small multiple of MATH does not affect the MATH's, but it affects MATH non-trivially. However, this perturbation is not localized, so it is not suitable for our purposes (we can't apply the globalization argument). Instead, let MATH be a polynomial of degree MATH in the coordinates MATH and their complex conjugates, such that MATH coincides with MATH up to order two at MATH. Note that the coefficients of MATH are bounded by uniform constants, and that its antiholomorphic part is at most of the order MATH (because MATH and MATH are asymptotically holomorphic); therefore, MATH is an admissible localized asymptotically holomorphic perturbation. Also, define MATH. Then one easily checks that the local section MATH of MATH corresponding to MATH and MATH is, at MATH, of the form MATH, where MATH is bounded from below. Moreover, let MATH and MATH : as above, this perturbation affects MATH and not the other MATH's, and we get that the corresponding local sections of MATH are of the form MATH, where the coefficient MATH is in MATH-th position. Once again, these perturbations satisfy both REF . Therefore, expressing MATH as a linear combination of MATH, REF yields transversality to MATH over a small ball centered at MATH by the usual argument. Now that both possible cases have been handled, we can apply the standard globalization argument to obtain uniform transversality to the stratum MATH. This gives properties MATH and MATH of REF . Now that all required transversality properties have been obtained, we perform further perturbations in order to achieve the other conditions in REF . These new perturbations are bounded by a fixed multiple of MATH, so the transversality properties are not affected. The argument is almost the same as in the case of REF (see REF); the adaptation to the higher-dimensional case is very easy. One first defines a suitable almost-complex structure MATH, by the same argument as in REF (except that one also considers the points of MATH and MATH besides the cusps). As explained in REF, a suitable perturbation makes it possible to obtain the local holomorphicity of MATH near these points, which yields conditions MATH, MATH and MATH ; the argument is the same in all three cases. Next, a generically chosen small perturbation yields the self-transversality of MATH (property MATH). Finally, as described in REF, a suitable perturbation yields property MATH along the branch curve without modifying MATH and MATH and without affecting the other compatibility properties. This completes the proof of the existence statement in REF . Uniqueness. The uniqueness statement is obtained by showing that, provided that MATH is large enough, the whole argument extends to the case of families of sections depending continuously on a parameter MATH. Then, given two sequences of quasiholomorphic maps, one can start with a one-parameter family of sections interpolating between them in a trivial way and perturb it in such a way that the required properties hold for all parameter values (with the exception of MATH when a node cancellation occurs). If one moreover checks that the construction can be performed in such a way that the two end points of the one-parameter family are not affected by the perturbation, the isotopy result becomes an immediate corollary. Observe that, in the one-parameter construction, the almost-complex structure is allowed to depend on MATH. Most of the above argument extends to REF families in a straightforward manner, exactly as in the four-dimensional case ; the key observation is that all the standard building blocks (existence of approximately holomorphic NAME coordinates MATH and of localized approximately holomorphic sections MATH, local transversality result, globalization principle, .) remain valid in the parametric case, even when the almost-complex structure depends on MATH. The only places where the argument differs from the case of REF are properties MATH, MATH and MATH, obtained in REF above. For property MATH, one easily checks that it is still possible in the parametric case to assume, after composing with suitable rotations depending continuously on the parameter MATH, that MATH and that MATH is bounded from below and directed along MATH. This makes it possible to define MATH and MATH as in the non-parametric case, and the parametric version of REF yields a suitable perturbation depending continuously on MATH. The argument of REF also extends to the parametric case, using the following observation. Fix a point MATH, and let MATH. For all values of MATH such that MATH is small enough (smaller than a fixed constant MATH), we can perform the construction as in the non-parametric case, defining MATH and MATH. If MATH is small enough (smaller than MATH), then we can apply the same argument as in the non-parametric case to define polynomials MATH and local sections MATH of MATH. However the definition of MATH needs to be modified as follows. Although it is still possible after a suitable rotation depending continuously on MATH to assume that MATH is small, the choice of an index MATH such that MATH lies far from the span of the MATH may depend on MATH. Instead, we define MATH as a unit vector in MATH depending continuously on MATH and such that MATH lies far from the span of MATH, and let MATH. Then the required properties are satisfied, and we can proceed with the argument. So, provided that MATH and MATH are both smaller than MATH, we can use REF to obtain a localized perturbation MATH depending continuously on MATH and such that MATH satisfies the desired transversality property near MATH. In order to obtain a well-defined perturbation for all values of MATH, we introduce a continuous cut-off function MATH which equals MATH over MATH and vanishes outside of MATH. Then, we set MATH, which is well-defined for all MATH and depends continuously on MATH. Since MATH coincides with MATH when MATH and MATH are smaller than MATH, the required transversality holds for these values of MATH ; moreover, for the other values of MATH we know that REF-jet of MATH already lies at distance more than MATH from the stratum MATH, and we can safely assume that MATH is much smaller than MATH, so the perturbation does not affect transversality. Therefore we obtain a well-defined local perturbation for all MATH, and the one-parameter version of the result of REF follows by the standard globalization argument. The argument of REF is extended to one-parameter families in the same way : given a point MATH, the same ideas as for REF yield, for all values of the parameter MATH such that REF-jet of MATH at MATH lies close to the stratum MATH, small localized perturbations MATH depending continuously on MATH and such that MATH satisfies the desired property over a small ball centered at MATH. As seen above, two different types of formulas for MATH arise depending on which component of the stratum MATH is being hit; however, the result of REF implies that, in any interval of parameter values such that the jet of MATH remains close to MATH, only one of the two components of MATH has to be considered, so MATH indeed depends continuously on MATH. The same type of cut-off argument as for REF then makes it possible to extend the definition of MATH to all parameter values and complete the proof. |
math/0007130 | The smoothness and symplecticity properties of the various submanifolds appearing in the statement follow from the observation made by NAME in CITE that the zero sets of approximately holomorphic sections satisfying a uniform transversality property are smooth and approximately MATH-holomorphic, and therefore symplectic. In particular, the smoothness and symplecticity of the fibers of MATH away from MATH follow immediately from REF : since MATH is bounded from below away from MATH (because it satisfies a uniform transversality property), and since the sections MATH are asymptotically holomorphic, it is easy to check that the level sets of MATH are, away from MATH, smooth symplectic submanifolds. NAME near the singular points is an immediate consequence of the local models MATH and MATH that we will obtain later in the proof. The corresponding properties of MATH and MATH are obtained by the same argument : MATH and MATH are the zero sets of asymptotically holomorphic sections, both satisfying a uniform transversality property (by REF and MATH of REF , respectively), so they are smooth and symplectic. We now study the local models at critical points of MATH. We start with the case of a cusp point MATH. By REF , MATH has complex rank MATH at MATH, so we can find local complex coordinates MATH on MATH near MATH such that MATH is the MATH axis. Pulling back MATH via the map MATH, we obtain, using REF, a MATH-holomorphic function whose differential does not vanish near MATH ; therefore, we can find a MATH-holomorphic coordinate chart MATH on MATH at MATH such that MATH. In the chosen coordinates, we get MATH, where MATH is holomorphic and MATH. Since MATH is by assumption a cusp point, the tangent direction to MATH at MATH lies in the kernel of MATH, that is, in the span of the MATH first coordinate axes ; after a suitable rotation we may assume that MATH is the MATH axis. Near the origin, MATH is characterized by its MATH components MATH, and the critical curve MATH is the set of points where these quantities vanish. Therefore, at the origin, MATH. Nevertheless, MATH vanishes transversely to MATH at the origin, so the matrix of second derivatives MATH, MATH, MATH, is non-degenerate (invertible) at the origin. In particular, the first column of MATH (corresponding to MATH) is non-zero, and therefore MATH is necessarily non-zero ; after a suitable rescaling of the coordinates we may assume that this coefficient is equal to MATH. Moreover, the invertibility of MATH implies that the submatrix MATH, MATH is also invertible, that is, it represents a non-degenerate quadratic form. NAME this quadratic form, we can assume after a suitable linear change of coordinates that the diagonal coefficients of MATH are equal to MATH and the others are zero. Therefore MATH is of the form MATH. Changing coordinates on MATH to replace MATH by MATH for all MATH, and on MATH to replace MATH by MATH, we can ensure that MATH. Observe that MATH is described near the origin by expressing the coordinates MATH as functions of MATH. By assumption the expressions of MATH are all of the form MATH. Substituting into the formula for MATH, and letting MATH, we get that local equations of MATH near the origin are MATH for MATH, and MATH. It follows that MATH is locally given in terms of MATH by the map MATH. Therefore, the transverse vanishing of MATH at the origin implies that MATH, so after a suitable rescaling we may assume that the coefficient of MATH in the power series expansion of MATH is equal to one. On the other hand, suitable coordinate changes can be used to kill all other degree MATH terms in the expansion of MATH : if MATH the coefficient of MATH can be made zero by replacing MATH by MATH ; similarly for MATH (replace MATH by MATH), MATH and MATH (replace MATH by MATH). So we get that MATH. It is then a standard result of singularity theory that the higher order terms can be absorbed by suitable coordinate changes. We now turn to the case of where MATH is a point of MATH which does not lie close to any of the cusp points. REF and MATH imply that the differential of MATH at MATH has real rank MATH and that its image lies close to a complex line in the tangent plane to MATH at MATH. Therefore, there exist local approximately holomorphic coordinates MATH on MATH such that MATH is the MATH axis. Moreover, because MATH is an approximately holomorphic function whose derivative at MATH satisfies a uniform lower bound, it remains possible to find local approximately holomorphic coordinates MATH on MATH such that MATH. As before, we can write MATH, where MATH is an approximately holomorphic function such that MATH. By REF restricts to MATH as an immersion at MATH, so the projection to the MATH axis of MATH is non-trivial. In fact, REF implies that, if MATH is very small at MATH, then a cusp point lies nearby ; so we can assume that the MATH component of MATH is larger than some fixed constant. As a consequence, one can show that MATH is locally given by equations of the form MATH, where the functions MATH are approximately holomorphic and have bounded derivatives. Therefore, a suitable change of coordinates on MATH makes it possible to assume that MATH is locally given by the equations MATH. Similarly, a suitable approximately holomorphic change of coordinates on MATH makes it possible to assume that MATH is locally given by the equation MATH. As a consequence, we have that MATH and, since the image of MATH at a point of MATH coincides with the tangent space to MATH, MATH vanishes at all points of MATH. In particular this implies that MATH for all MATH. Moreover, REF implies that MATH vanishes transversely at the origin, and therefore that the matrix MATH, MATH is invertible, that is, it represents a non-degenerate quadratic form. This quadratic form can be diagonalized by a suitable change of coordinates ; because the transversality property MATH is uniform, the coefficients are bounded between fixed constants. After a suitable rescaling, we can therefore assume that MATH is equal to MATH if MATH and MATH otherwise. In conclusion, we get that MATH, where MATH is the sum of a holomorphic function which vanishes up to order MATH at the origin and of a non-holomorphic function which vanishes up to order MATH at the origin and has derivatives bounded by MATH. Let MATH be the column vector MATH, and denote by MATH the vector MATH. Using the fact that MATH vanishes up to order MATH along MATH, we conclude that there exist matrix-valued functions MATH, MATH and MATH with the following properties : MATH ; (MATH and MATH are symmetric) ; MATH is approximately holomorphic and has uniformly bounded derivatives ; MATH ; MATH and MATH and their derivatives are bounded by fixed multiples of MATH. The implicit function theorem then makes it possible to construct a MATH approximately holomorphic change of coordinates of the form MATH (with MATH orthogonal, MATH approximately holomorphic, MATH), such that MATH becomes of the form MATH. Unfortunately, smooth coordinate changes are not sufficient to further simplify this expression; instead, in order to obtain the desired local model one must use as coordinate change an ``approximately holomorphic homeomorphism", which is smooth away from MATH but admits only directional derivatives at the points of MATH. More precisely, starting from MATH and using that MATH is bounded by MATH, we can write MATH . This gives the desired local model and ends the proof. |
math/0007130 | We only give a sketch of the proof of REF . As usual, we need to obtain two types of properties : uniform transversality conditions, which we ensure in the first part of the argument, and compatibility conditions, which are obtained by a subsequent perturbation. As in previous arguments, the various uniform transversality properties are obtained successively, using the fact that, because transversality is an open condition, it is preserved by any sufficiently small subsequent perturbations. The first transversality properties to be obtained are those appearing in REF , that is, the transversality to MATH of MATH for all MATH ; this easy case is for example, covered by the main result of CITE. One next turns to the transversality conditions arising from the requirement that the three sections MATH define quasiholomorphic maps from MATH to MATH : it follows immediately from the proof of REF that these properties can be obtained by suitable small perturbations. Next, we try to modify MATH, MATH and MATH in order to ensure that the restrictions to MATH of these three sections satisfy the transversality properties of REF . A general strategy to handle this kind of situation is to use the following remark REF : if MATH is a section of a vector bundle MATH over MATH, satisfying a uniform transversality property, and if MATH, then the uniform transversality to MATH over MATH of a section MATH of a vector bundle MATH is equivalent to the uniform transversality to MATH over MATH of the section MATH of MATH, up to a change in transversality estimates. This makes it possible to replace all transversality properties to be satisfied over submanifolds of MATH by transversality properties to be satisfied over MATH itself ; each property can then be ensured by the standard type of argument, using the globalization principle to combine suitably chosen local perturbations (see CITE for more details). However, in our case the situation is significantly simplified by the fact that, no matter how we perturb the sections MATH, MATH and MATH, the submanifold MATH itself is not affected. Moreover, the geometry of MATH is controlled by the transversality properties obtained on MATH ; for example, a suitable choice of the constant MATH (independent of MATH) ensures that the intersection of MATH with any ball of MATH-radius MATH centered at one of its points is topologically a ball (see for example, REF). Therefore, we can actually imitate all steps of the argument used to prove REF , working with sections of MATH over MATH. The localized reference sections of MATH over MATH that we use in the arguments are now chosen to be the restrictions to MATH of the localized sections MATH of MATH over MATH ; similarly, the approximately holomorphic local coordinates over MATH in which we work are obtained as the restrictions to MATH of local coordinate functions on MATH. With these two differences understood, we can still construct localized perturbations by the same algorithms as in REF and, using the standard globalization argument, achieve the desired transversality properties over MATH. Moreover, all these local perturbations are obtained as products of the localized reference sections by polynomial functions of the local coordinates. Therefore, they naturally arise as restrictions to MATH of localized sections of MATH over MATH, and so we actually obtain well-defined perturbations of the sections MATH, MATH and MATH over MATH which yield the desired transversality properties over MATH. We can continue similarly by induction on the dimension, until we obtain the transversality properties required of MATH, MATH and MATH over MATH, and finally the transversality properties required of MATH and MATH over MATH. Observe that, even though the perturbations performed over each MATH result in modifications of the submanifolds MATH REF lying inside them, these perturbations preserve the transversality properties of MATH, and so the submanifolds MATH retain their smoothness and symplecticity properties. We now turn to the second part of the argument, that is, obtaining the desired compatibility conditions. First observe that the proof of REF shows how, by a perturbation of MATH, MATH and MATH smaller than MATH, we can ensure that the various compatibility properties of REF are satisfied by the MATH-valued map MATH defined by these three sections. Next, we proceed to perturb MATH over a neighborhood of its ramification curve MATH, in order to obtain the required compatibility properties for MATH, but without losing those previously achieved for MATH near its ramification curve MATH. For this purpose, we first show that the curve MATH satisfies a uniform transversality property with respect to the hypersurface MATH in MATH. The only way in which MATH can fail to be uniformly transverse to MATH is if MATH becomes small at a point of MATH near MATH. Because MATH satisfies property MATH in REF , this can only happen if a cusp point or a tangency point of MATH lies close to MATH. However, REF implies that this point cannot belong to MATH. Therefore, two of the intersection points of MATH with MATH must lie close to each other. Observe that the points of MATH are precisely the critical points of the NAME pencil induced on MATH by MATH and MATH, that is, the tangency points of the map MATH. The transversality properties already obtained for MATH imply that two tangency points cannot lie close to each other ; we get a contradiction, so the cusps and tangencies of MATH must lie far away from MATH, and MATH and MATH are mutually transverse. This implies in particular that a small perturbation of MATH, MATH and MATH localized near MATH cannot affect properties MATH and MATH for MATH, and also that the only place where perturbing MATH might affect MATH is near the tangency points of MATH. We now consider the set MATH of points where we need to ensure properties MATH, MATH and MATH for MATH. The first step is as usual to perturb MATH into an almost-complex structure which is integrable near these points ; once this is done, we perturb MATH to make it locally holomorphic with respect to this almost-complex structure. We start by considering a point MATH, where the issue of preserving properties of MATH does not arise. We follow the argument in REF. First, it is possible to perturb the almost-complex structure MATH over a neighborhood of MATH in MATH in order to obtain an almost-complex structure MATH which differs from MATH by MATH and is integrable over a small ball centered at MATH. Recall from CITE that MATH is obtained by choosing approximately holomorphic coordinates on MATH and using them to pull back the standard complex structure of MATH ; a cut-off function is used to splice MATH with this locally defined integrable structure. Since we can choose the local coordinates in such a way that a local equation of MATH is MATH, we can easily ensure that MATH is, over a small neighborhood of MATH, a MATH-holomorphic submanifold of MATH. Next, we can perturb the sections MATH of MATH by MATH in order to make the projective map defined by them MATH-holomorphic over a neighborhood of MATH in MATH (see CITE). This holomorphicity property remains true for the restrictions to the locally MATH-holomorphic submanifold MATH. So, we have obtained the desired compatibility property near MATH. We now consider the case of a point MATH, where we need to obtain property MATH for MATH while preserving property MATH for MATH. We first observe that, by the construction of the previous step (getting property MATH for MATH at MATH), we have a readily available almost-complex structure MATH integrable over a neighborhood of MATH in MATH. In particular, by construction MATH is locally MATH-holomorphic and MATH is locally a MATH-holomorphic submanifold of MATH. We next try to make the projective map MATH holomorphic over a neighborhood of MATH, using once again the argument of CITE. The key observation here is that, because one of the sections MATH and MATH is bounded from below at MATH, we can reduce to a MATH-valued map whose first component is already holomorphic. Therefore, the perturbation process described in CITE only affects MATH, while the two other sections are preserved. This means that we can ensure the local MATH-holomorphicity of MATH without affecting MATH. It is easy to combine the various localized perturbations performed near each point of MATH ; this yields properties MATH, MATH and MATH of REF for MATH. We now use a generically chosen small perturbation of MATH, MATH and MATH in order to ensure property MATH, that is, the self-transversality of the critical curve of MATH. It is important to observe that, because MATH satisfies property MATH, the images by the projective map MATH of the points of MATH are all distinct from each other, and because MATH satisfies property MATH they are also distinct from MATH. Therefore, we can choose a perturbation which vanishes identically over a neighborhood of MATH ; this makes it possible to obtain property MATH for MATH without losing any property of MATH. Finally, by the process described in REF we construct a perturbation yielding property MATH along the critical curve of MATH ; this perturbation is originally defined only for the restrictions to MATH but it can easily be extended outside of MATH by using a cut-off function. The two important properties of this perturbation are the following : first, it vanishes identically near the points where MATH has already been made MATH-holomorphic, and in particular near the points of MATH ; therefore, none of the properties of MATH are affected, and properties MATH, MATH and MATH of MATH are not affected either. Secondly, this perturbation does not modify the critical curve of MATH nor its image, so property MATH is preserved. We have therefore obtained all desired properties for MATH. We can continue similarly by induction on the dimension, until all required compatibility properties are satisfied. Observe that, because the ramification curve of MATH remains away from its fiber at infinity MATH, we do not need to worry about the possible effects on MATH of perturbations of MATH. Therefore, the argument remains the same at each step, and we can complete the proof of the existence statement in REF in this way. The proof of the uniqueness statement relies, as usual, on the extension of the whole construction to one-parameter families ; this is easily done by following the same ideas as in previous arguments. |
math/0007132 | We use local coordinates. If MATH then MATH and the associated hamiltonian vector field is MATH . This expression shows that MATH projects to MATH, On the other hand, if MATH, one computes: MATH so the result follows. |
math/0007132 | If the rank of MATH at MATH is MATH we are done, so we can assume MATH. If MATH we proceed, by induction, straightening out vector fields of the form MATH. So let MATH and assume we have constructed coordinates MATH valid on a domain MATH, and a basis of sections for MATH over MATH, MATH such that MATH where MATH depend only on the MATH's. Since MATH, there exists a MATH such that the vector field MATH does not vanish at MATH. By relabeling, we can assume that MATH and we set MATH. By straightening out MATH, we can perform a change of coordinates MATH such that MATH . Replacing MATH by MATH, we see that we can assume MATH. Therefore, MATH where MATH. Using MATH for MATH, we see that MATH where the structure functions are related by MATH . We can think of this equation as a time-dependent linear o.d.e. for MATH in the variable MATH. Let us denote by MATH the fundamental matrix of solutions such that MATH, and by MATH its inverse. We consider new sections MATH . Then we find MATH . We conclude that there exist coordinates MATH and sections MATH, as in the statement of the theorem, such that REF hold, for some smooth functions MATH depending only on the MATH's. Since at MATH the bundle map MATH has rank MATH, we must have MATH. Comparing coefficients of MATH in MATH we check easily that the structure functions MATH must vanish for MATH. Using the NAME identity, we find for MATH and MATH, MATH . On the other hand, MATH so REF follows. |
math/0007132 | If MATH and MATH, there exists a piece-wise smooth path made of orbits of vector fields of the form MATH, with MATH a section of MATH. Integrating sections we can map MATH to MATH, so we may assume that these points of intersection are actually the same. Around MATH we choose coordinates MATH and sections MATH as in the Local Splitting Theorem. We interpolate between MATH and MATH by a family of manifolds MATH defined by equations of the form MATH . Then we look for a time-dependent section MATH which, by integration, gives a NAME algebroid automorphism MATH, covering a diffeomorphism MATH, which maps a neighborhood of MATH in MATH onto a neighborhood of MATH. Let us write MATH. In order for the MATH to track the MATH we must have the equations MATH satisfied along MATH. It is clear than one can choose MATH such that this equations holds. Integration of MATH gives a NAME algebroid automorphism MATH which induces a NAME algebroid isomorphism between MATH and MATH. |
math/0007132 | We compute using REF : MATH so we have MATH which shows that REF is satisfied. |
math/0007132 | By a result of CITE, a generalized distribution associated with a vector subspace MATH is integrable iff it is involutive and rank invariant. Taking MATH, so that MATH, REF shows that MATH is involutive iff the curvature REF-section vanishes. Hence, all it remains to show is that if the curvature vanishes and MATH is an integral curve of MATH then MATH is constant, for all small enough MATH. Fix MATH and let MATH be the flow MATH, let MATH be the flow of MATH and let MATH be the flow of MATH. We have MATH (see REF ). If MATH we claim that MATH for small enough MATH. In fact, the infinitesimal version of this relation is MATH which holds, since we are assuming that the curvature vanishes. Therefore, the flow MATH gives an isomorphism between MATH and MATH, for small enough MATH, so MATH is rank invariant. |
math/0007132 | From REF for the curvature and REF of the exterior MATH-derivative, we compute: MATH where the symbol MATH denotes cyclic sum over the subscripts. The first and fourth term vanish because of NAME 's identity, while the two middle terms cancel out. |
math/0007132 | Let MATH be a domain of a chart MATH where there exists a basis of trivializing sections MATH. On MATH, we define a linear MATH-connection by MATH and a MATH-connection on MATH by MATH where MATH and MATH denote, as usual, the structure functions for this choice of coordinates and basis. A straight forward computation shows that the relation MATH implies that MATH so MATH is a linear connection in MATH compatible with the NAME algebroid structure. If we take an open cover of MATH by such chart domains and if MATH is a partition of unity subordinated to this cover, then MATH and MATH define MATH-connections that satisfy MATH, that is, MATH is a connection in MATH compatible with the NAME algebroid structure. |
math/0007132 | A straightforward computation. |
math/0007132 | Let MATH be a MATH-path in MATH. We can find a time-dependent section MATH of MATH over MATH such that MATH. Using the notation above, we define a time-dependent section MATH over the tubular neighborhood such that for MATH and MATH . The lifts MATH are the integral curves of the vector field MATH defined by MATH so MATH is the map induced by the time-REF flow of MATH on MATH. The flow of MATH is induced by REF-parameter family of NAME algebroid homomorphisms MATH of MATH obtained by integrating the family MATH (see REF). The homomorphisms MATH gives a NAME algebroid isomorphism MATH which covers MATH. Relation REF follows since we have just shown that MATH is the time-REF map of some flow. |
math/0007132 | Recall that any piecewise smooth path MATH can be made into an NAME. By REF , it is enough to show that for every MATH there exists a neighborhood MATH of MATH in MATH such that if MATH is a piecewise smooth loop based at MATH and MATH is a piecewise smooth family with MATH then MATH is a inner automorphism of MATH. We use the same notation as in the proof of REF , so we construct a time-dependent section MATH in a tubular neighborhood of MATH which decomposes as MATH, and MATH is obtained by integrating this section up to time REF. It is clear that the parallel component MATH has no effect on the holonomy. Hence we can assume that MATH, MATH, MATH is a constant path and MATH. But then MATH is a REF-parameter family of automorphisms of MATH with MATH, so we conclude that MATH is an inner automorphism of MATH. |
math/0007132 | Assume first that MATH has trivial reduced holonomy and fix a base point MATH. We choose an embedding of MATH in MATH, a complementary subbundle MATH and trivialization so we can define the holonomy map MATH. Also, we choose a Riemannian metric on MATH. By compactness of MATH, there exists a number MATH such that every point MATH can be connected to MATH by a smooth MATH-path of length MATH. For some inner product on MATH, let MATH be the disk of radius MATH centered at MATH. For each MATH, we can choose a neighborhood MATH such that: CASE: for any piecewise-smooth MATH-path in MATH, starting at MATH, with length MATH and for any MATH, there exists a lifting with initial point MATH; CASE: the lifting of any MATH-loop based at MATH with initial point MATH has end point in MATH; CASE: MATH is invariant under all inner automorphisms of MATH; In fact, let MATH be MATH-loops such that their base loops MATH are generators of MATH, and let MATH be NAME algebroid automorphisms which represent the germs MATH. Since the reduced holonomy is trivial, there is a neighborhood MATH of MATH in MATH such that MATH, and MATH, for all i. Since MATH is transversally stable, we can choose a smaller neighborhood MATH invariant under all inner automorphisms. Given MATH and a MATH-path MATH connecting MATH to MATH, let us denote by MATH the diffeomorphism defined by lifting. It follows from REF above that if MATH is a MATH-path homotopic to MATH then MATH. It follows from REF that MATH is also invariant under all inner automorphisms. Let MATH be a neighborhood of MATH in MATH. There exists MATH such that for the corresponding MATH we have MATH. By compactness of MATH, we can choose MATH (independent of MATH) such that for the corresponding MATH we have MATH . Set MATH . Then MATH is a open neighborhood of MATH which is invariant under all inner automorphisms of MATH. Therefore, MATH is stable. If MATH are two points in the same leaf of MATH such that MATH, then there is a path MATH in this leaf connecting these two points. We can choose a loop MATH in MATH based at MATH such that MATH is a horizontal lift of this loop. Thus MATH is the image of MATH by MATH which is a inner automorphism of MATH. Therefore, MATH and MATH lie in the same leaf of MATH. We conclude that each leaf of MATH near MATH is a bundle over MATH with fiber a leaf of the transverse NAME algebroid structure. Assume now that MATH has finite reduced holonomy. We let MATH be a finite covering space such that MATH. If we embed MATH into MATH as above, and let MATH be the pull back bundle of MATH over MATH, we have a unique NAME algebroid structure MATH over MATH and a NAME algebroid homomorphism MATH which covers the natural map MATH. Moreover, the reduced holonomy of MATH along MATH is trivial, so we can apply the above argument to MATH and the theorem follows. |
math/0007132 | To check that REF is independent of the extensions considered, we fix a local basis of sections MATH for MATH in a neighborhood of MATH. If we write MATH for some functions MATH and MATH, we compute MATH . This expression shows that MATH only depends on the value of MATH at MATH and the values of MATH along MATH, that is, MATH and MATH. Relation REF also shows that MATH is in the kernel of MATH and so is a section of MATH. |
math/0007132 | In fact, let us see that the compatible connection MATH constructed in the proof of REF is a basic connection. We use the same notation as in that proof, so if MATH is a leaf of MATH and MATH, we write MATH and we have MATH . Therefore, for any REF-form MATH, we get MATH since MATH. It follows that for any REF-form MATH we have MATH . Similarly, for the connection MATH, we have MATH so if MATH we find MATH, and it follows that MATH . Since MATH we conclude that MATH defines a basic connection. |
math/0007132 | If MATH is any basic connection and MATH, we have MATH, so REF for the curvature tensor, gives MATH . But the right hand side is zero, because of the NAME identity. Similarly, if MATH is a differential form such that MATH, we have MATH. Hence, using MATH and the well known formula for the NAME derivative of the NAME bracket of vector fields, we find MATH so the second relation also holds. |
math/0007132 | We compute MATH where we have used first the linearity and symmetry of MATH, then the MATH-invariance of MATH, and last the NAME identity. |
math/0007132 | Suppose we have two MATH-connections in MATH with connection REF-sections MATH and MATH, and denote by MATH and MATH the MATH-sections they define through REF . We construct a REF-parameter family of connections with connection REF-section MATH, MATH, and we denote by MATH its curvature REF-section. By the transformation REF for the local connection REF-sections, the difference MATH is a MATH-valued REF-section, and we get a well defined MATH-section MATH by setting MATH where MATH, and the sum is over all permutations in MATH. We claim that MATH so MATH. To prove REF , we note that if we differentiate the structure REF we obtain MATH . Hence, using NAME 's identity, we have MATH so the claim follows. |
math/0007132 | Choose a MATH-connection in MATH which is induced by some covariant connection. Given MATH, this covariant connection gives a closed MATH-form MATH defined by a formula analogous to REF , and which induces the usual NAME homomorphism MATH. We check easily that MATH so the proposition follows. |
math/0007132 | According to REF we have MATH and we claim that MATH if MATH is odd (these are the vanishing primary classes that we mentioned to above). The proof that MATH is standard: we can choose an orthonormal basis of sections for MATH so that the curvature REF-sections take there values in MATH. But if MATH, we have MATH for any elementary symmetric function, since MATH is odd. Hence we obtain MATH. Consider now the connection MATH. Given MATH we choose local coordinates MATH around MATH and a basis of sections MATH as in the Local Splitting Theorem. Then MATH form a basis for MATH, and for the canonical skew-symmetric bilinear form MATH given by REF the only non-vanishing pairs are: MATH where MATH. Since MATH is induced by a basic connection, it is compatible with the NAME algebroid structure so from REF we conclude: MATH . On the other hand, from REF we find MATH . It follows that MATH is represented in the basis MATH by a matrix of the form: MATH with MATH a MATH symplectic matrix. Now, if MATH is any matrix of this form, it is clear that MATH, where MATH is the same as MATH with MATH, that is, MATH is symplectic. But if MATH is symplectic, we have MATH for any elementary symmetric function, since MATH is odd. Hence MATH. |
math/0007132 | Let MATH and MATH (respectively, MATH and MATH) be basic connections (respectively, riemannian connections). It follows from REF that MATH . Hence, it is enough to show that the cohomology classes of MATH and MATH are trivial. Consider first the basic connections MATH and MATH. The linear combination MATH is also a basic connection. If MATH, we fix splitting coordinates MATH around MATH and sections MATH as in the proof of REF . Then we see that, with respect to the basis MATH, the matrix representations of MATH and MATH are of the form REF . Hence, we conclude that if MATH, with MATH odd, MATH . Therefore, MATH, whenever MATH and MATH are basic connections. Now consider the riemannian connections MATH and MATH. The linear combination MATH is also a riemannian connection. All these connections are induced from covariant riemannian connections MATH, MATH and MATH, and we can define a differential form MATH of degree MATH by a formula analogous to REF . Moreover, this form is closed (because MATH is odd), and MATH. It follows from the homotopy invariance of MATH, using a suspension argument, as in the usual theory of secondary characteristic classes of foliations (see CITE, page REF), that MATH . Hence, the cohomology class MATH vanishes and so does the class MATH. |
math/0007132 | Choose a basic connection MATH and a riemannian connection MATH relative to some metric on MATH. We consider the transverse measure MATH to MATH associated with this metric. We claim that MATH so REF follows. Observe that it is enough to show that REF holds on the regular points of MATH, since the set of regular points is an open dense set and both sides are smooth sections in MATH. So assume that MATH is a regular point where MATH, and pick coordinates MATH around MATH and a basis of sections MATH as in the Local Splitting Theorem. Then MATH is given locally by: MATH where MATH is the matrix of inner products formed by elements in MATH. As in the proofs of the previous section, one computes the trace of the operator MATH relative to the basis MATH to be MATH . Also, since MATH is a metric connection, we find: MATH . So we conclude that: MATH . On the other hand, a straight forward computation using REF and the various relations in the Local Splitting Theorem at a regular point, shows that MATH . Comparing REF gives MATH so relation REF holds and the theorem follows. |
math/0007134 | The proof of REF is straightforward. Also, the claim that MATH is a metric on MATH follows immediately form REF and the equality MATH . Therefore we include only the proof of that equality: If MATH then MATH . Hence we can assume that MATH . Since MATH where MATH can by transformed into MATH by MATH elementary operations. Therefore, MATH . In order to prove the opposite inequality we need the following fact: If MATH is obtained from MATH by MATH elementary operations then MATH can be presented as MATH . We prove this fact by induction. The statement is true for MATH . Suppose it is also true for MATH and suppose that MATH is obtained from MATH by MATH elementary operations. By the inductive assumption, after MATH operations we get a product of MATH conjugates, MATH . The word MATH is obtained by inserting a word MATH into MATH . Therefore MATH for some MATH such that MATH . Thus MATH is a product of MATH conjugates. Since MATH is obtained from MATH by MATH elementary operations, MATH may be obtained from MATH also by MATH elementary operations. Therefore, by the fact proved above, MATH can be presented as a product of MATH conjugates of MATH's. Thus MATH . |
math/0007134 | CASE: We prove MATH first. If MATH has no connection then MATH and the inequality is obvious. Therefore we may assume that MATH has a connection MATH . Consider two kinds of operations on MATH: CASE: If there is an unconnected letter MATH in MATH then we delete this letter and we obtain a new word MATH with a connection MATH (composed of the same arcs as MATH). CASE: If there is a pair MATH of letters in MATH connected by an arc which is not nested (that is, MATH are neighbors) then we remove these letters and the arc connecting them and we obtain a new word MATH of a shorter length with a connection MATH . Observe that each of the above operations decreases the length of MATH and we can always apply at least one of them to MATH unless MATH is the trivial word MATH . Therefore any word MATH can be reduced to MATH by a sequence of operations of the first and the second type. Observe that Operation I changes MATH by at most one and decreases the norm of the connection on MATH by MATH . Operation II does not change MATH nor the norm of the connection on MATH . Since at the end of the process (when MATH) both MATH and the norm of the connection on MATH are MATH . Since this inequality holds for any connection MATH on MATH we also have MATH . CASE: We also claim that MATH . We can assume that MATH since otherwise MATH . There is a word MATH representing the same element of the group MATH as the word MATH with MATH. Each factor MATH has a connection of a norm MATH (a nested family of arcs). Therefore the product MATH has a connection of a norm MATH. The word MATH can be transformed into the word MATH by a sequence of insertions and deletions of subwords of the form MATH . Observe that after each insertion we obtain a new word with a connection of the same norm. Moreover, it is not difficult to see, that after each deletion we obtain a new word with a connection of the same norm, if MATH or lower or equal norm, if MATH . Therefore MATH has a connection MATH of norm MATH and, hence, MATH . |
math/0007136 | The sum MATH is congruent, modulo MATH, to the total number of edges in MATH consisting of a black vertex MATH and white vertex MATH or MATH. Given a loop MATH in MATH, the number of edges in it consisting of a black vertex MATH and white vertex MATH or MATH is equal to the number of times the loop crosses a ray with its end slightly below the deleted black vertex and with diagonal direction with respect to the square grid. (See REF .) This number is even if the deleted black vertex is in the exterior of the loop, and odd if it is in the interior. The same holds for MATH and the deleted white vertex. Hence a loop has an even number of edges consisting of a black vertex MATH and white vertex MATH or MATH if and only if it has exactly one of the two deleted vertices in its interior. We conclude the proof by summing over all loops. |
math/0007136 | The determinant on the left side of REF can be obtained from the determinant on the right side by elementary column operations. |
math/0007136 | The base case MATH is trivial. The inductive step MATH follows directly from REF and from Lemma A (special case). The inductive step MATH requires some more work. MATH where MATH is the NAME polynomial of degree MATH, which has the property MATH. For a definition of MATH, consult CITE. The only property of MATH we need to know is that it is a polynomial of degree r whose leading coefficient is one. We will adopt, for the sake of convenience, the convention MATH. We can now continue: MATH by Lemma A (special case). |
math/0007136 | The special case followed from the fact that the same sequence of column operations transforms the row MATH into MATH and the row MATH into MATH, for any values MATH,MATH. Therefore the same sequence transforms the row MATH into MATH. |
math/0007136 | MATH implies MATH . If we take the value at MATH, we obtain MATH . Similarly, MATH implies MATH . If we let MATH be the operator taking MATH to MATH, we can write MATH . Clearly the coefficient of MATH in MATH is equal to MATH times the coefficient of MATH in MATH which is equal to MATH times the coefficient of MATH in MATH that is, MATH. Hence MATH . Therefore MATH . |
math/0007139 | Reduction of the generic matrix MATH modulo MATH in REF leads to a generic remainder which depends on the parameters MATH. Moreover, since a NAME basis of MATH is parameter-free, this generic remainder has the property that its specialization to a fixed choice of parameters MATH gives the remainder of MATH modulo MATH. Thus setting the remainder to zero in REF corresponds to deriving conditions on the parameters MATH which makes the endomorphism given by MATH equal to the identity on MATH. This is possible if and only if MATH is a direct summand of MATH. The analogous statement holds for reduction of MATH modulo MATH and setting its resulting remainder to zero. Here, setting a remainder to zero is equivalent to the vanishing of the coefficients of its standard monomials, and we collect these vanishing conditions in the ideal MATH of MATH. Now a linear combination MATH is an isomorphism with inverse MATH if and only if the composition MATH is congruent to MATH modulo MATH and the opposite composition MATH is congruent to MATH modulo MATH. Thus the common zeroes MATH of MATH correspond to isomorphisms MATH and their inverses MATH. In particular, if MATH is the entire ring, which we detect by searching for MATH in a NAME basis of MATH, then there are no isomorphisms. On the other hand if MATH is proper, then MATH and MATH are isomorphic and we obtain an explicit isomorphism from finding any common solution of MATH. By REF , the invertible homomorphisms from MATH to MATH are NAME dense in the vector space MATH. Hence, a common solution can be explicitly found by by intersecting the zero locus of MATH with a suitable number of generic hyperplanes MATH. Because of denseness, each of these hyperplanes can be found in a finite number of steps. In other words, if MATH is proper, then there are only finitely many MATH for which the sum MATH is the unit ideal. |
math/0007139 | As we have seen, MATH is isomorphic to the nonempty affine variety MATH defined in the variables MATH. Here, a point of MATH with coordinates MATH corresponds to the isomorphism MATH. Now any isomorphism MATH induces an isomorphism from the variety to itself, sending MATH to MATH where MATH. This action is regular in MATH (since we showed MATH is rational in MATH), and transitive since MATH equals MATH. It follows that MATH is a smooth variety because it is a homogeneous space over itself via a transitive action. As we have seen in REF , MATH is NAME open in MATH (which is an affine space and therefore normal) and hence connected if MATH is algebraically closed. |
math/0007139 | A point MATH corresponds to an isomorphism with inverse MATH if and only if the system MATH has exactly one solution for MATH. This is equivalent to the MATH matrix MATH having rank MATH and the augmented matrix MATH also having rank MATH. The matrix MATH will have rank MATH if and only if any one of its MATH minors is nonzero. Similarly, the augmented matrix MATH will also have rank MATH if in addition all MATH minors vanish. We claim that each MATH minor of MATH must actually be identically zero. Otherwise it would impose an algebraic condition which must be satisfied by the isomorphisms in the coordinates MATH of MATH. But this cannot happen since the isomorphisms are an open set by REF . Thus we have shown that the space of non-isomorphisms MATH is defined by the equations obtained from the vanishing of all MATH minors of MATH. |
math/0007140 | Let MATH. Because MATH is an isolated singularity the exponents at MATH are all equal to MATH. Equivalently, there exists an orthonormal basis of MATH with respect to which the matrix of MATH is the direct sum of MATH copies of MATH. Thus, if MATH, then MATH is even. Let MATH be a MATH-invariant symmetric polynomial of degree MATH. Then, by a result of CITE (see also CITE) we have MATH where MATH is the Pfaffian (see CITE or CITE) and MATH is the closed MATH-form on MATH which represents the cohomology class induced by the NAME morphism applied to MATH via the NAME connection of MATH (see CITE or CITE). Note that the right hand side of REF is zero if MATH. It is easy to prove that MATH . By taking MATH, from REF, we obtain MATH . By taking MATH , from REF and the NAME Theorem we obtain that the NAME number of MATH is equal to the cardinal of MATH (this also follows from the NAME theorem or from CITE). But, by REF, the cardinal of MATH must be even and hence the NAME number of MATH is even. By definition, if MATH is not divisible by four then all the NAME numbers of MATH are zero. Suppose that MATH and let MATH be a partition of MATH. Denote by MATH the MATH-invariant symmetric polynomial of degree MATH such that MATH represents the MATH'th NAME class of MATH. Let MATH and recall that MATH is the direct sum of MATH copies of MATH. Then it is obvious that for any MATH-invariant symmetric polynomial MATH we have MATH where MATH is a constant which depends just on MATH and MATH but not on MATH. By taking MATH in REF it follows from REF that all the NAME numbers of MATH are zero. The fact that MATH has zero signature follows from NAME 's Signature Theorem (see CITE). |
math/0007140 | From REF it follows that we can write MATH where MATH and recall that MATH is a section of MATH (see CITE). In fact MATH is the nowhere zero function on MATH characterised by MATH then REF becomes MATH where MATH - note that with respect to a suitably chosen adapted orthonormal frame we have MATH. Also note that MATH is an almost complex structure on the normal bundle MATH (see CITE for the general case). Because MATH is a totally-geodesic submanifold of MATH we have that MATH; hence, from REF it follows that MATH . By the same reason we also have that MATH is the curvature form of the connection induced by MATH on MATH. The proof now follows from the NAME theorem. |
math/0007140 | Let MATH be the manifold (endowed with a circle action) obtained by blowing-up the isolated fixed points, that is, the points of MATH. Then, since signatures add when taking connected sums, the signature MATH of MATH is given by MATH . Because the exponents of each isolated fixed point are equal by hypothesis, the induced circle action on MATH has no isolated fixed points. This follows from REF (see also REF ). Thus we can apply CITE to obtain that MATH has signature zero. Combining this with REF gives MATH that is, the first equality of REF. Now, take a metric on MATH with respect to which MATH acts by isometries. Then by applying the NAME formula (see CITE) to the infinitesimal generator of this action we obtain MATH . Since MATH for MATH (because MATH if and only if MATH, MATH and MATH), we have MATH. By using this fact and REF becomes MATH . But by NAME theorem MATH which together with REF gives MATH which immediately yields the second equality of REF. |
math/0007140 | Because MATH and MATH are oriented, MATH is orientable. Thus we can choose MATH such that MATH . Clearly, MATH is smooth on MATH. Furthermore, when MATH, since MATH as we approach a critical point, MATH extends to a continuous vector field on MATH whose zero set is MATH and the flow of MATH extends to a continuous flow on MATH whose fixed point set is MATH . For any MATH, it is easy to see that MATH is a proper submersion. Then by a well-known result of CITE MATH restricted to MATH is the projection of a locally trivial fibre bundle. In particular, the orbit space MATH of MATH is a smooth manifold. Thus, MATH can be factorised as MATH where MATH has connected fibres and MATH is a covering projection. Let MATH be the unique metric on MATH with respect to which MATH becomes a Riemannian covering. It is obvious that MATH is a submersive harmonic morphism with compact connected fibres. From CITE it follows that MATH is the projection of a circle bundle where the action on the total space MATH is induced by the flow of MATH. |
math/0007140 | If MATH then, by REF , there exists a free MATH action on MATH whose orbits are connected components of the fibres of MATH. Hence, by the NAME theorem the NAME number of MATH is zero. Also, as is well-known (immediate consequence of REF), all the NAME numbers of MATH are zero. Suppose that MATH. If the set of critical points MATH is empty then the same argument as above implies that the NAME number and the NAME number of MATH are zero. Suppose that MATH and let MATH and MATH. By REF we can assume that MATH has connected fibres. Let MATH be a neighbourhood of MATH such that MATH and which is diffeomorphic to the closed ball of radius two centred at zero in MATH. Then MATH is a four-dimensional submanifold-with-boundary of MATH such that MATH. Furthermore, by REF , MATH is the projection of a MATH-bundle over MATH. Because MATH is the cone over MATH, MATH is the cone over its boundary. It easily follows (compare CITE) that the boundary of MATH must be simply-connected. Consider the embedding MATH. Let MATH be the NAME number of the MATH bundle MATH. Then, if MATH, MATH and, in particular, the fundamental group of MATH is MATH. (If MATH the bundle MATH is trivial.) But, MATH is diffeomorphic to the boundary of MATH which we have seen is simply-connected. It follows that MATH and, thus, we can suppose that MATH is smoothly equivalent to the projection of the cylinder of the NAME bundle MATH. Thus, by taking, if necessary, the equivariant connected sum of MATH and MATH, about MATH and MATH, where MATH is considered with its canonical circle action, we can suppose that on MATH we have a smooth circle action having MATH as a fixed point outside of which the action is free. By repeating this procedure about each point of MATH we obtain on MATH a smooth circle action whose fixed point set is MATH outside which the action is free. By REF , the NAME number of MATH is zero and its NAME number is even and equal to the cardinal of MATH . |
math/0007140 | Choose one of the orientations of MATH and let MATH be the corresponding volume-form with respect to MATH. Let MATH be the NAME number of MATH. By the NAME Theorem we have MATH where MATH is the NAME tensor of MATH and MATH , MATH are its self-dual and anti-self-dual components, respectively (see CITE). By REF , MATH and hence MATH. Thus, if MATH is half-conformally flat then it is conformally flat. Now, recall that, by the NAME Theorem, the NAME number of MATH is given by (see CITE, CITE): MATH where MATH is the scalar curvature of MATH and MATH is the trace-free part of the NAME tensor of MATH. If MATH is half-conformal flat and its scalar curvature is zero then REF becomes MATH . But, by REF , MATH and hence MATH must be NAME. If MATH is NAME and half-conformally flat then MATH has constant sectional curvature MATH (see CITE). By CITE and CITE we cannot have MATH. If MATH then, up to homotheties, the universal cover of MATH is MATH, a situation which cannot occur (see CITE). Hence MATH must be flat. If MATH is NAME and NAME is submersive then, by REF , the NAME number of MATH is zero. Thus, REF implies that MATH is flat. If MATH is NAME then, as a consequence of CITE, MATH must be submersive, since on a compact NAME manifold any Killing vector field is parallel (see CITE). Then, by REF , MATH. Now, REF implies that MATH is flat. The last assertion follows from CITE and an argument as in the proof of CITE. |
math/0007140 | Let MATH be a regular point of MATH and MATH. Let MATH be a unit vector. Because MATH is of constant curvature, by CITE, there exists an open neighbourhood MATH of MATH and a submersive harmonic morphism MATH with values in some NAME surface MATH such that its fibre through MATH is tangent to MATH. Then for any other unit vector MATH we can compose MATH with an isometry to obtain a submersive harmonic morphism MATH whose fibre through MATH is tangent to MATH. Then, MATH is a submersive harmonic morphism from an orientable NAME four-manifold to a NAME surface. By CITE, there exists an (integrable) Hermitian structure MATH on MATH with respect to which MATH is holomorphic. By restricting, if necessary, the family of MATH to an open subset of the unit sphere in MATH we can suppose that all the MATH induce the same orientation MATH on MATH. By the Riemannian NAME Theorem (see CITE) MATH is degenerate. Now, by a result of CITE either MATH or there is just exactly one pair MATH of (oriented) complex structures compatible with MATH. But the latter cannot occur because, obviously, if MATH then MATH. Hence MATH and the proof follows. |
math/0007140 | If MATH is submersive then the proof follows from CITE. Suppose that MATH has critical points. Then, by CITE, MATH has positive scalar curvature and there exists a smooth Killing vector field MATH tangent to the fibres of MATH whose zero set is equal to the set of critical points of MATH. Suppose that MATH is NAME. Its first NAME class is positive. Then, by CITE or CITE, MATH is either MATH or is obtained from MATH by blowing-up MATH distinct points, MATH; such a blow-up has signature MATH. But, by REF , MATH has signature zero and hence either MATH is MATH or is obtained from MATH by blowing-up one point. In the latter case, MATH is biholomorphic to MATH; but this admits no NAME metric (see CITE). If MATH then, by CITE, MATH is homothetic to MATH. Hence we may suppose that MATH is isometric to MATH. Let MATH be a critical point of MATH (note that by REF , in this case, MATH must have exactly four critical points). But MATH is also an isolated zero of MATH. Because MATH is a Killing vector field, MATH preserves each of the summands in the orthogonal decomposition MATH. Now, recall that MATH is an orthogonal complex structure on MATH. Hence we can suppose that MATH. It follows that, by composing MATH with the inverse of the map MATH, given by stereographic projection on each factor, we get a harmonic morphism MATH where MATH . Since the stereographic projection is conformal MATH is induced by the canonical circle action on MATH; note that this is an isometric action with respect to MATH. Let MATH be its infinitesimal generator and let MATH be the dilation of MATH. Then, up to a multiplicative constant we have MATH (see CITE, CITE) and MATH must be equal to the horizontal component of MATH. However, it can be checked without difficulty that then MATH cannot be extended over MATH. We have thus proved that MATH cannot be NAME. Because MATH has signature zero, from the main result of CITE it follows that MATH is MATH with one point blown-up endowed with the Page metric. However, from the discusion in CITE it follows that none of the Killing vector fields of the Page metric has isolated fixed points which are isolated as singularities as well so that this case is not possible either. |
math/0007143 | Let MATH be the NAME closure of MATH, and note that the NAME form is also invariant under MATH. Replacing MATH by a finite-index subgroup, we may assume MATH is NAME connected. Let MATH be an NAME decomposition of MATH. Assume MATH and MATH. From REF, we see that MATH contains codimension-one subspaces of both MATH and MATH. (Note that this implies MATH is nontrivial.) This implies that MATH is reductive. (Because MATH is a unipotent subgroup that intersects MATH nontrivially (and MATH), it must be contained in MATH, so MATH. Similarly, MATH. Therefore MATH.) Then, since MATH contains a codimension-one subgroup of MATH, and since MATH, it follows that MATH is conjugate to either MATH or MATH. Because MATH is a nontrivial, connected, normal subgroup of MATH, we conclude that MATH is conjugate to either MATH or MATH. Because MATH (else MATH, which contradicts the fact that there is a NAME form on MATH), we see that MATH is conjugate to MATH. Assume MATH and MATH does not contain any nontrivial hyperbolic elements. The NAME subgroup of MATH must be compact, and the radical of MATH must be unipotent, so choose a compact MATH and a nontrivial unipotent subgroup MATH such that MATH. Replacing MATH by a conjugate, we may assume, without loss of generality, that MATH. Let us show, for every nonzero MATH, that MATH. From the NAME Lemma CITE, we know there exists MATH, such that MATH is hyperbolic (and nonzero). If MATH, this contradicts the fact that MATH does not contain nontrivial hyperbolic elements. Let MATH and MATH be subspaces of MATH as in REF. Because MATH for every nonzero MATH, we have MATH (see REF), so MATH (see REF) and MATH (see REF). Assume, for the moment, that MATH. Then MATH . This implies that there exist MATH and MATH, such that MATH, with MATH hyperbolic (and nonzero). This contradicts the fact that MATH has no nontrivial hyperbolic elements. We may now assume that MATH. For any nonzero MATH, we have MATH so MATH. Then, from REF , we conclude that MATH, so MATH acts irreducibly on MATH. This contradicts the fact that MATH is a codimension-one subspace of MATH that is normalized by MATH. Assume MATH. We may assume MATH is nontrivial (otherwise Conclusion REF holds). We must have MATH, so we conclude that MATH and MATH. Because MATH consists of hyperbolic elements, this implies that MATH acts diagonalizably on MATH, for every MATH. Therefore MATH is conjugate to MATH, and, hence, to MATH. |
math/0007143 | Let MATH be the NAME closure of MATH, and note that the NAME form is also invariant under MATH. Replacing MATH by a finite-index subgroup, we may assume MATH is NAME connected. Let MATH be an NAME decomposition of MATH. For each real root MATH of MATH (with respect to the NAME subalgebra MATH), let MATH be the corresponding root space, and let MATH and MATH be the natural projections. Fix a choice of simple real roots MATH and MATH of MATH, such that MATH and MATH (so the positive real roots are MATH, MATH, MATH, and MATH). Replacing MATH by a conjugate under the NAME group, we may assume MATH. From the classification of parabolic subgroups CITE, we know that the only proper parabolic subalgebras of MATH that contain MATH are MATH . Assume MATH contains nontrivial hyperbolic elements. Let MATH. Replacing MATH by a conjugate, we may assume MATH. Assume MATH. Assume MATH is reductive. We may assume MATH (if necessary, replace MATH with its conjugate under the NAME reflection corresponding to the root MATH). Then, from REF, we see that MATH contains a codimension-one subspace of MATH. (Note that this implies MATH is nontrivial.) Let MATH, so MATH is the NAME algebra of a maximal unipotent subgroup of MATH. (In fact, MATH is the image of MATH under the NAME reflection corresponding to the root MATH.) From the preceding paragraph, we know that MATH . Therefore, REF implies that MATH is conjugate (under MATH) to either MATH or MATH. It is easy to see that MATH is not conjugate to MATH. (See CITE for an explicit description of MATH. If MATH is even, then MATH, so MATH does not contain a codimension-one subspace of any MATH-dimensional root space, but MATH does contain a codimension-one subspace of MATH.) Therefore, we conclude that MATH is conjugate to MATH. Then, because MATH is a nontrivial, connected, normal subgroup of MATH, we conclude that MATH is conjugate to MATH. Assume MATH is not reductive. Let MATH be a maximal parabolic subgroup of MATH that contains MATH (see REF ). By replacing MATH and MATH with conjugate subgroups, we may assume that MATH contains the minimal parabolic subgroup MATH. Therefore, the classification of parabolic subalgebras REF implies that MATH is either MATH or MATH. Assume MATH. From REF, we see that MATH (and hence also MATH) contains codimension-one subspaces of MATH and MATH. Because MATH does not contain such a subspace of MATH, we conclude that MATH. Furthermore, because the intersection of MATH with each of these subspaces does have codimension one, we conclude that MATH has precisely the same intersection; therefore MATH. Hence MATH. We now have MATH so REF implies MATH . This contradicts the fact that MATH. Assume MATH. From REF, we see that MATH (and hence also MATH) contains a codimension-one subspace of MATH. Because neither MATH nor MATH contains such a subspace, this is a contradiction. Assume MATH. We may assume MATH (if necessary, replace MATH with its conjugate under the NAME reflection corresponding to the root MATH). From REF, we see that MATH contains a codimension-one subspace of MATH. Because any codimension-one subalgebra of a nilpotent NAME algebra must contain the commutator subalgebra, we conclude that MATH contains MATH. Then we have MATH so REF implies MATH . Similarly, we also have MATH. It is now easy to show that MATH for every real root MATH, so MATH. This contradicts the fact that MATH. Assume MATH contains a regular element of MATH. Replacing MATH by a conjugate under the NAME group, we may assume that MATH is the sum of the positive root spaces, with respect to MATH. Then, from REF, we see that MATH contains codimension-one subspaces of both MATH and MATH. Therefore, MATH contains codimension-one subspaces of MATH and MATH, so the argument of REF applies. Assume MATH does not contain nontrivial hyperbolic elements. The NAME subgroup of MATH must be compact, and the radical of MATH must be unipotent, so choose a compact MATH and a nontrivial unipotent subgroup MATH such that MATH. Choose subspaces MATH and MATH of MATH as in REF. Let MATH be a proper parabolic subgroup of MATH, such that MATH and MATH (see REF ). Replacing MATH and MATH by conjugates, we may assume, without loss of generality, that MATH contains the minimal parabolic subgroup MATH (so MATH). From the classification of parabolic subalgebras REF , we know that there are only three possibilities for MATH. We consider each of these possibilities separately. First, though, let us show that MATH . From the NAME Lemma CITE, we know there exists MATH, such that MATH is hyperbolic (and nonzero). If MATH, this contradicts the fact that MATH does not contain nontrivial hyperbolic elements. Assume MATH is a minimal parabolic subgroup of MATH. Assume MATH. Choose MATH, such that MATH, and let MATH. (So MATH, MATH, and MATH.) From REF, we know that MATH. Then, because MATH, we conclude, from REF, that MATH. Because MATH, we have MATH. Therefore, because MATH, we conclude, from REF, that MATH, so REF implies that MATH. In particular, we have MATH, so REF implies MATH. Therefore, we have MATH . Because MATH, we conclude that MATH. This contradicts the fact that MATH does not contain nontrivial hyperbolic elements. Assume MATH. Replacing MATH by a conjugate under MATH, we may assume MATH, so MATH. We have MATH, so MATH for every MATH. Thus, REF implies MATH. We have MATH so REF implies that MATH. Assume MATH, for some MATH. From the conclusion of the preceding paragraph, we know that MATH. Because MATH and MATH, this implies MATH, so MATH (see REF). In particular, MATH, so REF implies MATH . This contradicts the fact that MATH does not contain nontrivial hyperbolic elements. Assume MATH, for some MATH. From NAME REF, we may assume MATH. Because MATH has codimension MATH in MATH (see REF), which contains REF-dimensional subspace MATH, we have a contradiction. Assume MATH. (This argument is similar to NAME REF.) Because MATH, we know that MATH, so MATH (see REF). In particular, MATH, so REF implies MATH . This contradicts the fact that MATH does not contain nontrivial hyperbolic elements. Assume MATH. We may assume there exists MATH, such that MATH (otherwise, MATH, so REF applies). Note that, because MATH, we have MATH. Assume MATH. Choose MATH, such that MATH. Then MATH, and MATH is a nonzero element of MATH, so we see that MATH. Because every unipotent subgroup of MATH is abelian, we conclude that MATH acts trivially on MATH, which means MATH. This contradicts REF . Assume MATH. We may assume, furthermore, that MATH (otherwise, by replacing MATH with its conjugate under the NAME reflection corresponding to the root MATH, we could revert to REF ). Then, because MATH, we must have MATH. Thus, MATH. From REF, we have MATH so MATH . On the other hand, from REF, we know that MATH contains a codimension-one subspace of MATH, so MATH contains a codimension-one subspace of MATH. This is a contradiction. Assume MATH. Note that, because MATH, we have MATH. From REF, we have MATH . Assume there is some nonzero MATH, such that MATH. Replacing MATH by a conjugate (under MATH), we may assume MATH. Let MATH. Because MATH contains a codimension-one subspace of MATH (see REF), one of the following two subsubsubcases must apply. Assume there exists MATH, such that MATH. From REF, we have MATH. Then, because MATH is a nonzero element of MATH, we conclude that MATH . This contradicts the fact that MATH, being compact, has no nontrivial unipotent elements. Assume MATH. For MATH, we have MATH. Thus, there is some MATH, such that MATH is hyperbolic (and nonzero). On the other hand, from REF, we have MATH. This contradicts the fact that MATH does not contain nonzero hyperbolic elements. Assume MATH, for every nonzero MATH. Fix some nonzero MATH. Because MATH, we must have MATH (so MATH). Replacing MATH by a conjugate (under MATH), we may assume MATH. Also, we may assume MATH (otherwise, we could revert to NAME REF by replacing MATH with its conjugate under the NAME reflection corresponding to the root MATH). Let MATH. Because MATH and MATH centralize each other, we see that MATH is a two-dimensional subspace of MATH consisting entirely of hyperbolic elements. Because MATH contains a codimension-one subspace of MATH (see REF), and MATH (see REF), we see that MATH contains a codimension-one subspace of MATH, so MATH contains nontrivial hyperbolic elements. This contradicts the fact that MATH does not contain nontrivial hyperbolic elements. |
math/0007147 | Let MATH be the affine proalgebraic group whose existence is guaranteed by REF . The semisimplicity assumption, and the assumption that MATH has only finite number of irreducible representations imply that MATH is finite. |
math/0007147 | See for example, [ES, REF ]. |
math/0007147 | By REF , MATH as desired. |
math/0007147 | By a fundamental result of NAME and NAME REF , MATH and hence by REF , MATH is central. Now, we have MATH CITE, so MATH. This shows that MATH in every irreducible representation of MATH. But MATH and MATH are central, so they act as scalars in this representation, which proves REF . |
math/0007147 | By REF , MATH hence the result follows from REF . |
math/0007147 | Straightforward. |
math/0007147 | Let MATH be the functor defined by MATH . Let MATH and MATH where MATH is defined by MATH . Then it is straightforward to verify that MATH is a symmetric functor. |
math/0007147 | This is a special case of [DM, REF ]. This theorem states that the category of fiber functors from MATH to MATH an affine proalgebraic group, is equivalent to the category of MATH-torsors over MATH . But, MATH is algebraically closed, hence there exists only a unique MATH-torsor over MATH . |
math/0007147 | Let MATH . We have two fiber functors MATH arising from the forgetful functor; namely, the trivial one and the one defined by MATH respectively. By REF , MATH are isomorphic. Let MATH be an isomorphism. By definition, MATH is a family of MATH-linear isomorphisms. By naturality, the diagram MATH commutes for any two objects MATH and morphism MATH . In particular, for MATH the left regular representation, we get that MATH . Let MATH and MATH be the right multiplication by MATH . Then, MATH which is equivalent to saying that MATH is an isomorphism of the right regular representation. Hence, MATH for all MATH where MATH is invertible. Moreover, setting MATH and MATH in REF yields that MATH which is equivalent to MATH . Now, by REF , the diagram MATH commutes. In particular, MATH . Hence, MATH as desired. |
math/0007147 | Let MATH be an isomorphism of triangular NAME algebras. Then MATH defines an isomorphism of triangular NAME algebras from MATH to MATH . This implies that the element MATH is a symmetric twist for MATH. Thus, for some invertible MATH one has MATH. Let MATH. It is obvious that MATH is a NAME algebra isomorphism, so it comes from a group isomorphism MATH. We have MATH, as desired. |
math/0007147 | Let MATH be the category of finite-dimensional representations over MATH of MATH . This is a semisimple abelian MATH-linear category with finitely many irreducible objects, which has a structure of a symmetric rigid category (see REF). In this case, the categorical dimension MATH of MATH is equal to MATH . Since the NAME element MATH of MATH is MATH it follows that it is equal to the ordinary dimension of MATH as a vector space. In particular, all categorical dimensions are non-negative integers. In this situation we can apply REF . Let MATH be the finite group and functor corresponding to our category MATH . In particular, MATH preserves categorical dimensions of objects, and hence their ordinary dimensions. Thus, we may identify MATH as vector spaces functorially for all MATH . There exists an algebra isomorphism MATH such that for all object MATH the MATH-module structure on MATH is given via pull back along MATH . Let MATH be the set of all the isomorphism classes of irreducible representations of MATH . Since MATH is an equivalence of tensor rigid categories MATH is the set of all the isomorphism classes of irreducible representations of MATH . Since MATH and MATH are semisimple algebras we can fix algebra isomorphisms MATH and MATH . This determines an isomorphism of algebras MATH (of course, this isomorphism is not unique). Now, by the construction of MATH the vector space MATH is a MATH-module via pull back along MATH . By REF , there exists a family of natural MATH-module isomorphisms MATH indexed by all couples MATH . Consider MATH and set MATH . For all MATH and MATH . Consider the MATH-module maps MATH and MATH determined by MATH and MATH for all MATH respectively. By naturality, the diagram MATH commutes. In particular, MATH which is equivalent to MATH as desired. In particular, the isomorphism MATH is determined by MATH . Since MATH is in its image, it follows that MATH is invertible. Hence MATH is invertible as well. Set MATH . For all MATH . Since the map MATH is an isomorphism of MATH-modules, we have that MATH . By REF , this is equivalent to MATH . The claim follows now after replacing MATH by MATH . For all MATH . We first show that MATH . Let MATH denote the right unit constraints (we use the same notation for both categories for convenience). Then by REF , we have that the diagram MATH commutes. In particular, MATH which implies that MATH . Similarly, MATH . Now, since the map MATH is an isomorphism of MATH-modules, we have that MATH . By REF , this is equivalent to MATH . Write MATH . Then the last equation implies that MATH which in turn (since MATH is an isomorphism) implies that MATH and the result follows. MATH is a twist for MATH . Let MATH denote the associativity constraints in the categories MATH and MATH (we use the same notation for both categories for convenience). By REF , the diagram MATH commutes. In particular, MATH which is equivalent to MATH . Write MATH . Substitute MATH for MATH in the last equation, and use REF to get MATH . Since MATH is an isomorphism, this is equivalent to saying that MATH satisfies the first part of REF . Now, we already showed in the proof of REF that MATH . Thus, the second part of REF follows from REF after replacing MATH with MATH . By REF, MATH is a triangular semisimple NAME algebra, and the map MATH is an isomorphism of NAME algebras. Finally, let MATH denote the commutativity constrains in the categories MATH and MATH (again, we use the same notation for both categories for convenience). Recall that MATH is in particular a symmetric functor. MATH . By REF , the diagram MATH commutes. In particular, MATH . Therefore, MATH which is equivalent to the desired result. This completes the proof of the theorem. |
math/0007147 | Let MATH be the minimal triangular sub NAME algebra of MATH . By REF , there exist a finite group MATH and a minimal twist MATH for MATH such that MATH as triangular NAME algebras. We may as well assume that MATH . Let MATH be the inclusion map. Then MATH is an injective morphism of triangular NAME algebras as well. In particular, MATH which is equivalent to MATH . Moreover, since MATH is triangular, MATH is cocommutative. Therefore there exists a finite group MATH such that MATH . Hence, MATH . Since MATH is a sub NAME algebra of MATH is a subgroup of MATH . Finally, the uniqueness follows from REF . |
math/0007150 | Obviously MATH is a parallel frame for each MATH. So writing MATH one easily computes MATH and MATH. But MATH. |
math/0007150 | We define MATH and MATH. REF ensures that MATH and MATH are smooth at MATH and MATH. Using MATH this in turn implies that MATH has the form MATH for some MATH. Since the zeroes of MATH are fixed we know that MATH and MATH are constant. We write MATH. One gets MATH and MATH . This can be used to show MATH. Again REF ensures that MATH for some MATH and MATH. But then the integrability condition MATH gives up to a factor MATH and possible constant real parts MATH and MATH that MATH and MATH are fixed to be MATH and MATH. The additional term MATH in MATH corresponds to the (trivial) tangential flow which always can be added. The form MATH gives MATH and MATH. Thus one ends up with MATH. |
math/0007150 | Let MATH and MATH be the solutions to REF corresponding to MATH and MATH. One has MATH and MATH with MATH and MATH. The ansatz MATH leads to the compatability condition MATH or MATH which gives: MATH . Thus MATH and MATH are completely determined. To show that they give dressed solutions we note that since MATH the zeroes of MATH are the same as the ones of MATH (and the ones of MATH coincide with those of MATH). Therefore they do not depend on MATH and MATH. Moreover at these points the kernel of MATH coincides with the one of MATH. Thus it does not depend on MATH or MATH either. Now REF gives the desired result. |
math/0007150 | Obviously the above transformation coincides with the dressing described in the last section with MATH in REF . This proves the lemma. |
math/0007150 | One has MATH for fixed time MATH and MATH . |
math/0007150 | One can proof the theorem by direct calculations or using the equivalence of the dIHM model and the dNLSE stated in REF . If the curve MATH evolves by rigid motion its complex curvature may vary by a phase factor only: MATH or MATH. Plugging this in REF gives MATH which is equivalent to REF with MATH. |
math/0007150 | Analogous to the smooth case. |
math/0007150 | Literally the same as for REF . |
math/0007150 | Let us look at an elementary quadrilateral: For notational simplicity let us write MATH and MATH. If we denote the angles MATH and MATH with MATH and MATH we get MATH with MATH and MATH as in REF . MATH is the corresponding angle along the edge MATH. Note that MATH and MATH are coupled by MATH . To get an equation for MATH from this we need to have all vectors in one plane. So set MATH. Then conjugation with MATH is a rotation around MATH with angle MATH. If we replace MATH by MATH and MATH by MATH REF becomes quaternionic but stays valid (one can think of it as a complex equation with different MATH). REF now reads MATH . We can write this in homogenous coordinates: MATH carries a natural right MATH-modul structure, so one can identify a point in MATH with a quaternionic line in MATH by MATH. In this picture our equation gets MATH . Bringing MATH and MATH on the right hand side gives us finally the matrix MATH . Since we know that this map sends a sphere of radius MATH onto itself, we can project this sphere stereographically to get a complex matrix. The matrix MATH projects MATH onto MATH. Its inverse is given by MATH . One easily computes MATH with MATH. This completes our proof. |
math/0007150 | Comparing the orders in MATH on both sides in REF gives two equations MATH . The first holds trivially from construction the second gives MATH . This can be checked by elementary calculations using REF for the real part of MATH. |
math/0007150 | We use the notation from REF . Since MATH and since MATH is completely determined by MATH and MATH we have, that MATH. On the other hand on can determine MATH by MATH and MATH. Since MATH REF says that MATH and MATH must lie in MATH. |
math/0007150 | For the NAME transformation MATH look at the vector field MATH given by MATH. This must have a zero. |
math/0007150 | With notations as in REF we know MATH and MATH giving us MATH which proofs the claim since MATH goes to REF if MATH tends to MATH. |
math/0007150 | Evolving by rigid motion means for the complex curvature of a discrete curve, that it must stay constant up to a possible global phase, i. CASE: MATH. Due to REF the evolution equation for MATH reads MATH . Using MATH gives MATH and finally MATH . So the complex curvature of curves that move by rigid motion solve MATH with some real parameters MATH and MATH which clearly holds for discrete elastic curves. |
math/0007151 | Substituting MATH and MATH into REF one gets REF . A right-handed version of the Proposition also holds. |
math/0007151 | By uniqueness of the decomposition REF one can set MATH in an arbitrary basis MATH of MATH. Properties of the MATH-algebra map MATH as well as MATH are to be verified. |
math/0007151 | MATH is a MATH version of REF . Taking into account REF one calculates MATH . Hence MATH. Applying now MATH to the both sides, gives MATH. |
math/0007151 | It is enough to check REF on basis vectors: MATH due to REF and using the properties of the antipode. |
math/0007151 | Let MATH be any basis in MATH and MATH the dual basis in MATH. With respect to the given basis one can define the generalize derivations MATH as MATH . Thus MATH. Substituting MATH into REF and comparing the coefficients in front of the same basis vectors we conclude MATH which is equivalent to REF . |
math/0007158 | In the following the sums over MATH are taken over all pair partitions MATH of the set MATH and sums over MATH are taken over all subsets of the set MATH. Products over MATH are taken over MATH. From REF follows that for any MATH and indexes MATH we have: MATH and furthermore MATH . We define MATH REF shows that the corresponding summands in the definitions of MATH and MATH are equal unless there are some indexes MATH such that MATH. There are MATH choices of these indexes and again from REF we have MATH where MATH and therefore from REF we have MATH . We have MATH where measures MATH are defined on the set of all sequences MATH, MATH by REF . From REF follows that this sequence converges pointwise to the product measure defined on the atoms by MATH . Since measures MATH and the measure MATH are probabilistic, this convergence is uniform and the statement of the theorem follows. |
math/0007158 | We define MATH. In the following sums over MATH are taken over all pair partitions MATH of the set MATH with additional property that there exist MATH and MATH such that MATH. From REF we have that MATH where in the last inequality we used REF . |
math/0007158 | Our goal is to construct a sequence MATH such that MATH holds. However, MATH . The first summand converges to MATH by REF . From REF we have that MATH . From REF it follows that this expression converges to MATH. |
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