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The aim of this paper is to get some results about ruled surfaces which configure a projective theory of scrolls and ruled surfaces. Our ideas follow the viewpoint of Corrado Segre, but we employ the contemporaneous language of locally free sheaves. The results complete the exposition given by R. Hartshorne and they have not appeared before in the contemporaneous literature. | The Projective Theory of Ruled Surfaces | 12,300 |
The goal of this work is to pursue the study of pseudo-effective line bundles and vector bundles. Our first result is a generalization of the Hard Lefschetz theorem for cohomology with values in a pseudo-effective line bundle. The Lefschetz map is shown to be surjective when (and, in general, only when) the pseudo-effective line bundle is twisted by its multiplier ideal sheaf. This result has several geometric applications, e.g. to the study of compact K\"ahler manifolds with pseudo-effective canonical or anti-canonical line bundles. Another concern is to understand pseudo-effectivity in more algebraic terms. In this direction, we introduce the concept of an "almost" nef line bundle, and mean by this that the degree of the bundle is nonnegative on sufficiently generic curves. It can be shown that pseudo-effective line bundles are almost nef, and our hope is that the converse also holds true. This can be checked in some cases, e.g. for the canonical bundle of a projective 3-fold. From this, we derive some geometric properties of the Albanese map of compact K\"ahler 3-folds. | Pseudo-effective line bundles on compact Kähler manifolds | 12,301 |
In this paper we study the ample cone of the moduli space $\mgn$ of stable $n$-pointed curves of genus $g$. Our motivating conjecture is that a divisor on $\mgn$ is ample iff it has positive intersection with all 1-dimensional strata (the components of the locus of curves with at least $3g+n-2$ nodes). This translates into a simple conjectural description of the cone by linear inequalities, and, as all the 1-strata are rational, includes the conjecture that the Mori cone is polyhedral and generated by rational curves. Our main result is that the conjecture holds iff it holds for $g=0$. More precisely, there is a natural finite map $r: \vmgn 0. 2g+n. \to \mgn$ whose image is the locus $\rgn$ of curves with all components rational. Any 1-strata either lies in $\rgn$ or is numerically equivalent to a family $E$ of elliptic tails and we show that a divisor $D$ is nef iff $D \cdot E \geq 0$ and $r^*(D)$ is nef. We also give results on contractions (i.e. morphisms with connected fibers to projective varieties) of $\mgn$ for $g \geq 1$ showing that any fibration factors through a tautological one (given by forgetting points) and that the exceptional locus of any birational contraction is contained in the boundary. Finally, by more ad-hoc arguments, we prove the nefness of certain special classes. | Towards the ample cone of $\mgn$ | 12,302 |
We propose a definition of ``nonabelian mixed Hodge structure'' together with a construction associating to a smooth projective variety $X$ and to a nonabelian mixed Hodge structure $V$, the ``nonabelian cohomology of $X$ with coefficients in $V$'' which is a (pre-)nonabelian mixed Hodge structure denoted $H=Hom(X_M, V)$. We describe the basic definitions and then give some conjectures saying what is supposed to happen. At the end we compute an example: the case where $V$ has underlying homotopy type the complexified 2-sphere, and mixed Hodge structure coming from its identification with $\pp ^1$. For this example we show that $Hom (X_M,V)$ is a namhs for any smooth projective variety $X$. | Nonabelian mixed Hodge structures | 12,303 |
Local models are schemes defined in linear algebra terms that describe the 'etale local structure of integral models for Shimura varieties and other moduli spaces. We point out that the flatness conjecture of Rapoport-Zink on local models fails in the presence of ramification and that in that case one has to modify their definition. We study in detail certain modifications of the local models for G=R_{E/F}GL(n), with E/F a totally ramified extension, and for a maximal parahoric level subgroup. The special fibers of these models are subschemes of the affine Grassmannian. We show that the new local models are smoothly equivalent to "rank varieties" of matrices, are flat, normal, with rational singularities and that their special fibers contain the expected Schubert strata. A corollary is that Schubert varieties in the affine Grassmannian are smoothly equivalent to nilpotent orbit closures and are normal with rational singularities, even in positive characteristics. We give some applications to the calculation of sheaves of nearby cycles and describe a relation with geometric convolution. Finally, in the general EL case, we replace the flatness conjecture of Rapoport-Zink with a conjecture about the modified local models. | Local models in the ramified case I. The EL-case | 12,304 |
We prove that the fake projective planes constructed from the diadic discrete group discovered by Cartwright, Mantero, Steger, and Zappa are connected Shimura varieties associated to a certain unitary group. The necessary Shimura data, as well as the unitary group, are explicitly described. We also give a field of definition of these fake projective planes. | Arithmetic structure of CMSZ fake projective planes | 12,305 |
Monodromy in analytic families of smooth complex surfaces yields groups of isotopy classes of orientation preserving diffeomorphisms for each family member X. For all deformation classes of minimal elliptic surfaces with p_g>q=0, we determine the monodromy group of a representative X, i.e. the group of isometries of the intersection lattice L_X:=H_2/torsion generated by the monodromy action of all families containing X. To this end we construct families such that any isometry is in the group generated by their monodromies or does not respect the invariance of the canonical class or the spinor norm. | Monodromy groups of regular elliptic surfaces | 12,306 |
We prove two results on the defining ideals of certain varieties of matrices. Let us fix two positive integers r, e. Let M(r) be the set of r x r matrices over a field K. We consider the closed subscheme of the nilpotent variety of M(r) over K defined by the conditions char_A(T)=T^r, A^e=0. We prove that when the characteristic of K is zero this scheme is reduced. Also for e=2 we prove that this scheme is reduced over a field K of arbitrary characteristic. These results were motivated by the questions of G. Pappas and M. Rapoport (compare their paper ''Local models in the ramified case I. The EL-case", math.AG/0006222) and give answers to some of their conjectures. | Two results on equations of nilpotent orbits | 12,307 |
M. Nori proved that on a projective smooth variety, a bundle is finite, (that is the ring it generates has dimension 0), if and only if it trivializes on a finite cover. In this note, we consider bundles of degree 0 on an elliptic curve. We prove that the dimension of the ring generated by such a bundle is the same as the dimension of a torsor trivializing the bundle (and is 1). | Relation between the dimensions of the ring generated by a vector bundle
of degree zero on an elliptic curve and a torsor trivializing this bundle | 12,308 |
This is the Foreword to the book ``Explicit birational geometry of 3-folds'', edited by A. Corti and M. Reid, CUP Jun 2000, ISBN: 0 521 63641 8, with papers by K. Altmann, A. Corti, A.R. Iano-Fletcher, J. Koll\'ar, A.V. Pukhlikov and M. Reid. One of the main achievements of algebraic geometry over the last 20 years is the work of Mori and others extending minimal models and the Enriques--Kodaira classification to 3-folds. This book is an integrated suite of papers centred around applications of Mori theory to birational geometry. Four of the papers (those by Pukhlikov, Fletcher, Corti, and the long joint paper Corti, Pukhlikov and Reid) work out in detail the theory of birational rigidity of Fano 3-folds; these contributions work for the first time with a representative class of Fano varieties, 3-fold hypersurfaces in weighted projective space, and include an attractive introductory treatment and a wealth of detailed computation of special cases. | Explicit birational geometry of 3-folds, Foreword | 12,309 |
In this paper we give a description of hypersurfaces with trivial ring $AK(S)$, introduced by the second author as following. Let $X$ be an affine variety and let $G(X)$ be the group generated by all $\Bbb {C}^+$-actions on $X$. Then $AK(X)$ is the subring of all regular $G(X)-$ invariant functions on $X.$ We show that a smooth affine surface $S$ with $AK(S)=\Bbb C$ is quasihomogeneous and so may be obtained from a smooth rational projective surface by deleting a divisor of special form, which is called a ``zigzag''. We denote by $A$ the set of all such surfaces, and by $H$ those which have only three components in the zigzag. We prove that for a surface $S \in A$ the following statements are equivalent: 1. $S$ is isomorphic to a hypersurface; 2. $S$ is isomorphic to a hypersurface, defined by equation $xy=p(z)$ in $\Bbb {C}^3 ,$ where $p$ is a polynomial with simple roots only; 3. $S$ admits a fixed-point free $\Bbb {C}^+$- action; 4. $S\in H.$ Moreover, if $S_1 $ belongs to $H,$ and $S_2$ does not, then $S_1\times \Bbb {C}^k\not\cong S_2\times \Bbb {C}^k$ for any $k\in\Bbb N$. | Affine surfaces with $AK(S)=\Bbb C.$ | 12,310 |
A Brill-Noether locus is a subvariety of M_g consisting of curves having certain linear series g^r_d. We study the relative position of Brill-Noether loci with respect to the gonality stratification of M_g. We construct smooth curves in P^r of given degree and genus and having the `expected' gonality. As an application we give a new proof of our result about the Kodaira dimension of the moduli space of curves of genus 23. | Brill-Noether Loci and the Gonality Stratification of $M_g$ | 12,311 |
We present formulas for the homogenous multivariate resultant as a quotient of two determinants. They extend classical Macaulay formulas, and involve matrices of considerably smaller size, whose non zero entries include coefficients of the given polynomials and coefficients of their Bezoutian. These formulas can also be viewed as an explicit computation of the morphisms and the determinant of a resultant complex. | Explicit formulas for the multivariate resultant | 12,312 |
We prove that a variation of graded-polarizable mixed Hodge structure over a punctured disk with unipotent monodromy, has a limiting mixed Hodge structure at the puncture (i.e., it is admissible in the sense of [SZ]) which splits over $\R$, if and only if certain grading of the complexified weight filtration, depending smoothly on the Hodge filtration, extends across the puncture. In particular, the result exactly supplements Schmid's Theorem for pure structures, which holds for the graded variation, and gives a Hodge-theoretic condition for the relative monodromy weight filtration to exist. | Singularities of variations of mixed Hodge structure | 12,313 |
We determine the class of the Hilbert scheme of points on a surface in the Grothendieck group of varieties. As a corollary we obtain its class in the Grothendieck group of motives. We give some applications to moduli spaces of sheaves on surfaces. The paper is related to math/0005249 of de Cataldo and Migliorini. | On the motive of the Hilbert scheme of points on a surface | 12,314 |
The characteristic cycle of a complex of sheaves on a complex analytic space provides weak information about the complex; essentially, it yields the Euler characteristics of the hypercohomology of normal data to strata. We show how perverse cohomology actually allows one to extract the individual Betti numbers of the hypercohomology of normal data to strata, not merely the Euler characteristics. We apply this to the ``calculation'' of the vanishing cycles of a complex, and relate this to the work of Parusi\'nski and Brian\c{c}on, Maisonobe, and Merle on Thom's $a_f$ condition. | Perverse Cohomology and the Vanishing Index Theorem | 12,315 |
Recall that the moduli space of smooth (that is, stable) cubic curves is isomorphic to the quotient of the upper half plane by the group of fractional linear transformations with integer coefficients. We establish a similar result for stable cubic surfaces: the moduli space is biholomorphic to a quotient of the compex 4-ball by an explict arithmetic group generated by complex reflections. This identification gives interesting structural information on the moduli space and allows one to locate the points in complex hyperbolic 4-space corresponding to cubic surfaces with symmetry, e.g., the Fermat cubic surface. Related results, not quite as extensive, were announced in alg-geom/9709016. | The Complex Hyperbolic Geometry of the Moduli Space of Cubic Surfaces | 12,316 |
The Castelnuovo-Mumford regularity of varieties of degree r and dimension n in the r-dimensional projective space that have an extremal secant line, is at least d-r+n+1. We classify these varieties and show that their regularity is exactly d-r+n+1, as predicted by the regularity conjecture. | On the regularity of varieties having an extremal secant line | 12,317 |
New relations among the genus-zero Gromov-Witten invariants of a complex projective manifold $X$ are exhibited. When the cohomology of $X$ is generated by divisor classes and classes ``with vanishing one-point invariants,'' the relations determine many-point invariants in terms of one-point invariants. | New recursions for genus-zero Gromov-Witten invariants | 12,318 |
A simple corollary of the localization theorem (due to the author and, independently, to Lian-Liu-Yau) is applied to several problems in enumerative geometry. New formulas for Schubert calculus on flag manifolds, due to Kong, and a new reconstruction theorem for genus-zero Gromov-Witten invariants, due to Bertram-Kley, are discussed, as well as some simple functorial properties of Givental's J-function. This paper will appear in the issue of the Michigan Mathematical Journal dedicated to Bill Fulton. | Some applications of localization to enumerative problems | 12,319 |
We give a proof of generalizations of the classical Arakelov inequality valid for the degree $d$ of the relative canoincal bundle of a family of curves of genus $g$ over a complete curve of genus $p$ under the assumption that the monodromy around the singular fibers is unipotent. This relative canonical bundle is the (canonical extension of) the Hodge bundle and the inequality is generalized to the degrees of the Hodge bundles of a complex variation of Hodge structures. | Arakelov-type inequalities for Hodge bundles | 12,320 |
This paper surveys the authors recent work on two variable elliptic genus of singular varieties. The last section calculates a generating function for the elliptic genera of symmetric products. This generalizes the classical results of Macdonald and Zagier. | Elliptic Genera of singular varieties, orbifold elliptic genus and
chiral deRham complex | 12,321 |
In this paper, we introduce and study two new types of non-abelian zeta functions for curves over finite fields, which are defined by using (moduli spaces of) semi-stable vector bundles and non-stable bundles. A Riemann-Weil type hypothesis is formulated for zeta functions associated to semi-stable bundles, which we think is more canonical than the other one. All this is motivated by (and hence explains in a certain sense) our work on non-abelian zeta functions for number fields. | New Non-Abelian Zeta Functions for Curves over Finite Fields | 12,322 |
In this paper we show a Zariski pair of sextics which is not a degeneration of the original example given by Zariski. This is the first example of this kind known. The two curves of the pair have a trivial Alexander polynomial. The difference in the topology of their complements can only be detected via finer invariants or techniques. In our case the generic braid monodromies, the fundamental groups, the characteristic varieties and the existence of dihedral coverings of $\PP^2$ ramified along them can be used to distinguish the two sextics. Our intention is not only to use different methods and give a general description of them but also to bring together different perspectives of the same problem. | Sextics with singular points in special position | 12,323 |
In this paper, we show that the B\"acklund transformations of Painlev\'e equations come from birational maps of rational surfaces constructed by Okamoto as the spaces of initial conditions. The simultaneous resolutions of rational double points of the families of rational surfaces give rise to flops which are origins of symmetry groups of Painlev\'e equation. Moreover we point out that Kodaira--Spencer theory of deformations of rational surfaces explains a geomrtric meaning of Painlev\'e equations. | Painlevé equations and deformations of rational surfaces with rational
double points | 12,324 |
Given a tame Galois branched cover of curves pi: X -> Y with any finite Galois group G whose representations are rational, we compute the dimension of the (generalized) Prym variety corresponding to any irreducible representation \rho of G. This formula can be applied to the study of algebraic integrable systems using Lax pairs, in particular systems associated with Seiberg-Witten theory. However, the formula is much more general and its computation and proof are entirely algebraic. | Dimensions of Prym Varieties | 12,325 |
We present a complete list of extremal elliptic K3 surfaces. There are altogether 325 of them. The first 112 coincides with Miranda-Persson's list for semi-stable ones. The data include the transcendental lattice which determines uniquely the K3 surface by a result of Shioda and Inose, the singular fibre type and the Mordell Weil group. As an application, we give a sufficient condition for the topological fundamental group of complement to an ADE-configuration of smooth rational curves on a K3 surface to be trivial. | Classification of extremal elliptic K3 surfaces and fundamental groups
of open K3 surfaces | 12,326 |
We decompose the Marsden-Weinstein reductions for the moment map associated to representations of a quiver. The decomposition involves symmetric products of deformations of Kleinian singularities, as well as other terms. As a corollary we deduce that the Marsden-Weinstein reductions are irreducible varieties. | Decomposition of Marsden-Weinstein reductions for representations of
quivers | 12,327 |
We consider p-divisible groups (also called Barsotti-Tate groups) in characteristic p, their deformations, and we draw some conclusions. For such a group we can define its Newton polygon (abbreviated NP). This is invariant under isogeny. For an abelian variety (in characteristic p) the Newton polygon of its p-divisible group is ``symmetric''. In 1963 Manin conjectured that conversely any symmetric Newton polygon is ``algebroid''; i.e., it is the Newton polygon of an abelian variety. This conjecture was shown to be true and was proved with the help of the ``Honda-Serre-Tate theory''. We give another proof. Grothendieck showed that Newton polygons ``go up'' under specialization: no point of the Newton polygon of a closed fiber in a family is below the Newton polygon of the generic fiber. In 1970 Grothendieck conjectured the converse: any pair of comparable Newton polygons appear for the generic and special fiber of a family. This was extended by Koblitz in 1975 to a conjecture about a sequence of comparable Newton polygons. We prove these conjectures. | Newton polygons and formal groups: Conjectures by Manin and Grothendieck | 12,328 |
We prove that if X is a locally complete intersection variety, then X has all the jet schemes irreducible if and only if X has canonical singularities. After embedding X in a smooth variety Y, we use motivic integration to express the condition that X has irreducible jet schemes in terms of data coming from an embedded resolution of X in Y. We show that this condition is equivalent with having canonical singularities. In the appendix, this result is used to prove a generalization of Kostant's freeness theorem to the setting of jet schemes. | Jet Schemes of Locally Complete Intersection Canonical Singularities | 12,329 |
This paper generalizes for non-abelian theta functions a number of formulae valid for theta functions of Jacobian varieties. The addition formula, the relation with the Szego kernel and with the multicomponent KP hierarchy and the behavior under cyclic coverings are given. | Addition Formulae for Non-Abelian Theta Functions and Applications | 12,330 |
For a cubic surface X, by considering the intermediate Jacobian J(Y) of the triple covering Y of the 3-dimensional projective space branching along X, Allcock, Carlson and Toledo constructed a period map per from the family of marked cubic surfaces to the four dimensional complex ball embedded in the Siegel upper half space of degree 5. We give an expression of the inverse of per in terms of theta constants by constructing an isomorphism between J(Y) and a Prym variety of a cyclic 6-ple covering of the projective line branching at seven points. | Theta constants associated to cubic three folds | 12,331 |
Let $\psi$ be the period map for a family of the cyclic triple coverings of the complex projective line branching at six points. The symmetric group $S_6$ acts on this family and on its image under $\psi.$ In this paper, we give an $S_6$-equivariant expression of $\psi^{-1}$ in terms of fifteen theta constants. | Theta constants associated with the cyclic triple coverings of the
complex projective line branching at six points | 12,332 |
We study the birationality (onto its image) of the Abel-Prym morphism associated with a Prym-Tuyrin variety. We use such result to prove that Picard bundles over Prym varieties are simple and moreover they are stable when the Abel-Prym morphism is not birational. As a consequence we obtain that Picard bundles over moduli spaces of stable vector bundles with fixed determinant are simple. We prove that Picard bundles over moduli spaces of rank 2 vector bundles on curves of genus 2 bundles are stable. | On Picard bundles over Prym varieties | 12,333 |
We describe genus g>1 potentials of semisimple Frobenius structures. Our formula can be considered as a definition in the axiomatic context of Frobenius manifolds. In Gromov-Witten theory, it becomes a conjecture expressing higher genus GW-invariants in terms of genus 0 GW-invariants of symplectic manifolds with generically semisimple quantum cup-product. The conjecture is supported by the corresponding theorem about equivariant GW-invariants of tori actions with isolated fixed points. The parallel theory of gravitational descendents is also presented. | Semisimple Frobenius structures at higher genus | 12,334 |
To a Hodge structure V of weight k with CM by a field K we associate Hodge structures V_{-n/2} of weight k+n for n positive and, under certain circumstances, also for n negative. We show that these `half twists' come up naturally in the Kuga-Satake varieties of weight two Hodge structures with CM by an imaginary quadratic field. | Half twists of Hodge structures of CM-type | 12,335 |
In the 80's D. Eisenbud and J. Harris considered the following problem: ``What are the limits of Weierstrass points in families of curves degenerating to stable curves?'' But for the case of stable curves of compact type, treated by them, this problem remained wide open since then. In the present article, we propose a concrete approach to this problem, and give a quite explicit solution for stable curves with just two irreducible components meeting at points in general position. | Limit canonical systems on curves with two components | 12,336 |
We show that elliptic solutions of the classical Yang-Baxter equation can be obtained from triple Massey products on elliptic curve. We introduce the associative version of this equation which has two spectral parameters and construct its elliptic solutions. We also study some degenerations of these solutions. | Classical Yang-Baxter equation and the $A_{\infty}$-constraint | 12,337 |
We give necessary and sufficient conditions for a big and nef line bundle L of any degree on a K3 surface or Enriques surface to be k-very ample and k-spanned. Furthermore, we give necessary and sufficient conditions for a spanned and big line bundle on a K3 surface to be birationally k-very ample and birationally k-spanned, and relate these concepts to the Clifford index and gonality of smooth curves in |L|. | On kth-order embeddings of K3 surfaces and Enriques surfaces | 12,338 |
A Hodge structure V of weight k on which a CM field acts defines, under certain conditions, a Hodge structure of weight k-1, its half twist. In this paper we consider hypersurfaces in projective space with a cyclic automorphism which defines an action of a cyclotomic field on a Hodge substructure in the cohomology. We determine when the half twist exists and relate it to the geometry and moduli of the hypersurfaces. We use our results to prove the existence of a Kuga-Satake correspondance for certain cubic 4-folds. | Half twists and the cohomology of hypersurfaces | 12,339 |
In the 80's M. Cornalba and J. Harris discovered a relation among the Hodge class and the boundary classes in the Picard group with rational coefficients of the moduli space of stable, hyperelliptic curves. They proved the relation by computing degrees of the classes involved for suitable one-parameter families. In the present article we show that their relation can be obtained as the class of an appropriate, geometrically meaningful empty set, thus conforming with C. Faber's general philosophy to finding relations among tautological classes in the Chow ring of the moduli space of curves. The empty set we consider is the closure of the locus of smooth, hyperelliptic curves having a special ramification point. | A geometric interpretation and a new proof of a relation by Cornalba and
Harris | 12,340 |
We determine the complete list of anticanonically embedded quasi smooth log Fano 3-folds in weighted projective 4-spaces. This implies that the Reid-Fletcher list of 95 types of anticanonically embedded quasi smooth terminal Fano threefolds in weighted projective 4-spaces is complete. We also prove that there are only finitely many families of quasi smooth Calabi-Yau hypersurfaces in weighted projective spaces of any given dimension. | Fano hypersurfaces in weighted projective 4-spaces | 12,341 |
We prove a representability theorem for moduli functors of framed torsion-free sheaves on nonsingular complex projective surfaces, using formal geometry along a curve in the surface. This has as a consequence that a certain restriction morphism for moduli stacks of sheaves has fibers that are schemes. | Representability for some moduli stacks of framed sheaves | 12,342 |
Fix a ruled surface S obtained as the projective completion of a line bundle L on a complex elliptic curve; we study the moduli problem of parametrizing certain pairs consisting of a sheaf E on S and a map of E to a fixed reference sheaf on S. We prove that the full moduli stack for this problem is representable by a scheme in some cases. Moreover, the moduli stack admits an action by the group ${\mathbf C}^*$, and we determine its fixed-point set, which leads in some special cases to explicit formulas for the rational homology of the moduli space. | Moduli spaces of framed sheaves on certain ruled surfaces over elliptic
curves | 12,343 |
We describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori, in terms of structure of the zero- and one-dimensional orbits. The class of varieties to which our formula applies includes Schubert varieties in flag varieties and affine flag varieties. We also prove a monotonicity result on local intersection cohomology stalks. | From moment graphs to intersection cohomology | 12,344 |
Previous results on genera g of F_{q^2}-maximal curves are improved: (1) Either g\leq (q^2-q+4)/6, or g=\lfloor(q-1)^2/4\rfloor, or g=q(q-1)/2; (2) The hypothesis on the existence of a particular Weierstrass point in \cite{at} is proved; (3) For q\equiv 1\pmod{3}, q\ge 13, no F_{q^2}-maximal curve of genus (q-1)(q-2)/3 exists; (4) For q\equiv 2\pmod{3}, q\ge 11, the non-singular F_{q^2}-model of the plane curve of equation y^q+y=x^{(q+1)/3} is the unique F_{q^2}-maximal curve of genus g=(q-1)(q-2)/6; (5) Assume \dim(\cD_\cX)=5, and char(\fq)\geq 5. For q\equiv 1\pmod{4}, q\geq 17, the Fermat curve of equation x^{(q+1)/2}+y^{(q+1)/2}+1=0 is the unique F_{q^2}-maximal curve of genus g=(q-1)(q-3)/8. For q\equiv 3\pmod{4}, q\ge 19, there are exactly two F_{q^2}-maximal curves of genus g=(q-1)(q-3)/8, namely the above Fermat curve and the non-singular F_{q^2}-model of the plane curve of equation y^q+y=x^{(q+1)/4}. The above results provide some new evidences on maximal curves in connection with Castelnuovo's bound and Halphen's theorem, especially with extremal curves; see for instance the conjecture stated in Introduction. | On the genus of a maximal curve | 12,345 |
Smooth surfaces have finitely generated canonical rings and projective canonical models. For normal surfaces, however, the graded ring of multicanonical sections is possibly nonnoetherian, such that the corresponding homogeneous spectrum is noncompact. I construct a canonical compactification by adding finitely many non-Q-Gorenstein points at infinity, provided that each Weil divisor is numerically equivalent to a Q-Cartier divisor. Similar results hold for arbitrary Weil divisors instead of the canonical class. | What is missing in canonical models for proper normal algebraic
surfaces? | 12,346 |
We prove that a smooth hypersurface of degree >2 and dimension >1 admits no endomorphism of degree >1 (for hyperquadrics this is due to Paranjape and Srinivas). We then collect some general results on endomorphisms of projective manifolds; we prove in particular that ramified endomorphisms occur only on varieties of negative Kodaira dimension, and we classify those Del Pezzo surfaces which admit such endomorphism. | Endomorphisms of hypersurfaces and other manifolds | 12,347 |
We compute the transcendental part of the normal function corresponding to the Deligne class of a cycle in K_1 of a mirror family of quartic K3 surfaces. The resulting multivalued function does not satisfy the hypergeometric differential equation of the periods and we conclude that the cycle is indecomposable for most points in the mirror family. The occurring inhomogenous Picard-Fuchs equation are related to Painlev\'e VI type differential equations. | The transcendental part of the regulator map for K_1 on a mirror family
of K3 surfaces | 12,348 |
Let X be a smooth projective curve over a field of characteristic p>0. We show that the Hitchin morphism, which associates to a Higgs bundle its characteristic polynomial, has a non-trivial deformation over the affine line. This deformation is constructed by considering the moduli stack of t-connections on vector bundles on X and an analogue of the p-curvature, and by observing that the associated characteristic polynomial is, in a suitable sense, a p-th power. | On the Hitchin morphism in positive characteristic | 12,349 |
We show that a general plane curve of degree at least 4 is uniquely determined by the full set of its bitangent lines. This problem has an elementary solution for degree at least 5, and the paper is almost entirely devoted to curves of degree 4, where we generalize the result to nodal quartics. In other words, we show that a general curve of genus 3 can be recovered from its 28 odd theta-characteristics. | Recovering plane curves from their bitangents | 12,350 |
In this paper, we consider moduli spaces of stable sheaves on abelian surfaces. Our main assumption is the primitivity of the associated Mukai vector. We construct many isomorphisms of muduli spaces induced by Fourier-Mukai functor. As an application, we show that deformation type of these spaces are determined by their dimension. We next show that the fiber of the albanese map is irreducible symplectic manifolds In particular, we describe the period by using Mukai lattice. We also discuss deformation type of moduli spaces of stable sheaves on K3 surfaces. | Moduli spaces of stable sheaves on abelian surfaces | 12,351 |
Let $G$ be an algebraic group and let $\widetilde{\mathfrak g}$ be the corresponding affine algebra on some level. Consider the induced module $V:=Ind^{\widetilde{\mathfrak g}}_{{\mathfrak g}[[t]](O_{G[[t]]})$, where $O_{G[[t]]}$ is the ring of regular functions on the group $G[[t]]$. In this paper we show that $V$ is naturally a vertex operator algebra, which is "responsible" for D-modules on the loop group $G((t))$. Using the techiques of VOA we show that $V$ is in fact a bimodule over the affine algebra. In addition, we show that $V$ possesses a remarkable property related to its BRST reduction with respect to $\widetilde{\mathfrak g}$. This paper has a considerable intersection with a recent preprint of Gorbunov, Malikov and Schechtman. | Differential operators and the loop group via chiral algebras | 12,352 |
Let k be an algebraically closed field and X a smooth projective k-variety. A famous theorem of A. A. Roitman states that the canonical map from the degree zero part of the Chow group of zero cycles on X to the group of k-points of its Albanese variety induces an isomorphism on torsion prime to the characteristic of k. In the present paper we prove a generalisation to quasi-projective varieties admitting a smooth compactification. As was first observed by Ramachandran, for such a generalisation one should replace the Chow group of zero cycles by Suslin's 0-th algebraic singular homology group and the Albanese variety by the generalised Albanese of Serre. The method of proof is new even in the projective case and makes the motivic nature of the Albanese transparent. We also prove that the generalised Albanese map is an isomorphism if k is the algebraic closure of a finite field. | On the Albanese map for smooth quasi-projective varieties | 12,353 |
We show that a generic vector field on an affine space of positive characteristic admits an invariant algebraic hypersurface. This contrast with Jouanolou's Theorem that shows that in characteristic zero the situation is completely opposite. That is a generic vector field in the complex plane does not admit any invariant algebraic curve. | An analogous of Jouanolou's Theorem in positive characteristic | 12,354 |
A group action on a smooth variety provides it with the natural stratification by irreducible components of the fixed point sets of arbitrary subgroups. We show that the corresponding maximal wonderful blowup in the sense of MacPherson-Procesi has only abelian stabilizers. The result is inspired by the abelianization algorithm of Batyrev. | Wonderful blowups associated to group actions | 12,355 |
This paper contains some applications of Fourier-Mukai techniques to the birational geometry of threefolds. In particular, we prove that birational Calabi-Yau threefolds have equivalent derived categories. To do this we show how flops arise naturally as moduli spaces of perverse coherent sheaves. | Flops and derived categories | 12,356 |
Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard to handle, it has been shown in a previous paper that there exists a partial resolution of singularities which transforms a bundle of possibly badly singular curves into a bundle of nodal and cuspidal plane cubics. In cases which are of interest for classification theory, the total spaces of these bundles will clearly be projective. It is, however, generally false that an arbitrary bundle of plane cubics is globally projective. For that reason the question of projectivity seems to be of interest, and the present work gives a characterization of the projective bundles. | Projective bundles of singular plane cubics | 12,357 |
We study polystable Higgs bundles twisted by a line bundle over a compact K\"ahler manifold. These form a Tannakian category when the first and second Chern classes of the bundle are zero. In this paper we identify the corresponding Tannaka group in the case in which the line bundle is of finite order. This group is described in terms of the pro-reductive completion of the fundamental group of the manifold, and the character associated to the line bundle. | Twisted Higgs bundles and the fundamental group of compact Kähler
manifolds | 12,358 |
We give upper and lower bounds for the order of the top Chern class of the Hodge bundle on the moduli space of principally polarized abelian varieties. We also give a generalization to higher genera of the famous formula $12 \lambda_1=\delta$ for genus 1. | The top Chern class of the Hodge bundle on the moduli space of abelian
varieties | 12,359 |
Let M be a moduli space of stable sheaves on a K3 or Abelian surface S. We express the class of the diagonal in the cartesian square of M in terms of the Chern classes of a universal sheaf. Consequently, we obtain generators of the cohomology ring of M. When S is a K3 and M is the Hilbert scheme of length n subschemes, this set of generators is sufficiently small in the sense that there aren't any relations among them in the stable cohomology ring. When S is the cotangent bundle of a Riemann surface, we recover the result of T. Hausel and M. Thaddeus: The cohomology ring of the moduli spaces of Higgs bundles is generated by the universal classes. | Generators of the cohomology ring of moduli spaces of sheaves on
symplectic surfaces | 12,360 |
The class in the Brauer group of a quaternion algebra over a field is 2-torsion. We study the following question: Which 2-torsion elements of the Brauer group of a complex function field are representable by quaternion algebras? Using intersection theory to show that a certain cohomology class (on a smooth projective model) is the class of an algebraic cycle, we arrive at an obstruction, defined on a subgroup of the 2-torsion of the Brauer group, to representability by quaternion algebras. For the function fields of some complex threefolds, the obstruction map is computed and found to be nontrivial. | Hodge-theoretic obstruction to existence of quaternion algebras | 12,361 |
Smooth complex surfaces polarized with an ample and globally generated line bundle of degree three and four, such that the adjoint bundle is not globally generated, are considered. Scrolls of a vector bundle over a smooth curve are shown to be the only examples in degree three. Two classes of examples in degree four are presented, one of which is shown to characterize regular such pairs. A Reider-type theorem is obtained in which the assumption on the degree of the polarization is removed. | On polarized surfaces of low degree whose adjont bundles are not spanned | 12,362 |
We classify minimal pairs (X, G) for smooth rational projective surface X and finite group G of automorphisms on X. We also determine the fixed locus X^G and the quotient surface Y = X/G as well as the fundamental group of the smooth part of Y. The realization of each pair is included. Mori's extremal ray theory and recent results of Alexeev and also Ambro on the existence of good anti-canonical divisors are used. | Automorphisms of Finite Order on Rational Surfaces | 12,363 |
We give an explicit formula for the exponents (i.e. the spectra up to the shift by one) of an irreducible plane curve singularity in terms of Puiseux pairs. As an application we prove in this case Hertling's conjecture that the variance (i.e. the square of the standard deviation) of the exponents is bounded by the difference between the maximal and minimal exponents divided by 12. | Exponents of an irreducible plane curve singularity | 12,364 |
By proving a particular case of a conjecture of Drezet, we show that a component of the Maruyama scheme of the semi-stable sheaves on the projective space $\PP^n$ of rank n and Chern polynomial $(1+t)^{n+2}$ is isomorphic to the Kronecher moduli $N(n+1,2,n+2)$, for any odd n. In particular, such scheme defines a smooth minimal compactification of the moduli space of the rational normal curves in $\PP^n$, that generalizes the construction defined by G. Ellinsgrud, R. Piene and S. Str{\o}mme in the case $n=3$. | On the moduli space of the Schwarzenberger bundles | 12,365 |
Let R be a complete discrete valuation ring of mixed characteristics, with algebraically closed residue field k. We study the existence problem of equivariant liftings to R of Galois covers of nodal curves over k. Using formal geometry, we show that this problem is actually a local one. We apply this local-to-global principle to obtain new results concerning the existence of such liftings. | Relèvement galoisien des revêtements de courbes nodales | 12,366 |
Given a del Pezzo surface of degree d between 1 and 6, possibly with rational double points, we construct a "tautological" holomorphic G-bundle over X, where G is a reductive group which is an appropriate conformal form of the simply connected complex linear group whose coroot lattice is isomorphic to the primitive cohomology of the minimal resolution of X. For example, in case d=3 and X is a smooth cubic surface, the rank 27 vector bundle over X associated to the G-bundle constructed above and the standard 27-dimensional representation of E_6 is a direct sum of the line bundles associated to the 27 lines on X. We also discuss the restriction of the G-bundle to smooth hyperplane sections. | Exceptional groups and del Pezzo surfaces | 12,367 |
Let X be a smooth complex projective variety, and let Y in X be a smooth very ample hypersurface such that -K_Y is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) the "mirror formula", i.e. we show that the generating function of the genus zero 1-point Gromov-Witten invariants of Y can be obtained from that of X by a certain change of variables (the so-called "mirror transformation"). Moreover, we use the same techniques to give a similar expression for the (virtual) numbers of degree-d plane rational curves meeting a smooth cubic at one point with multiplicity 3d, which play a role in local mirror symmetry. | Relative Gromov-Witten invariants and the mirror formula | 12,368 |
Let $f:\Cal C\to S$ be a flat family of curves over a smooth curve $S$ such that $f$ is smooth over $S_0=S\ssm\{s_0\}$ and $f^{-1}(s_0)=\Cal C_0$ is irreducible with one node. We have an associated family $\Cal M_{S_0}\to S_0$ of moduli spaces of semistable vector bundles and the relative theta line bundle $\Theta_{S_0}$. We are interested in the problem: to find suitable degeneration $\Cal M_S$ of moduli spaces and extension $\Theta_S$ of theta line bundles such that the direct image of $\Theta_S$ is a vector bundle on $S$ with a logarithmic projective connection. In this paper, we figured out the conditions of existence of the connection and solved the problem for rank one. | Logarithmic heat projective operators | 12,369 |
In this article we study the inverse of the period map for the family $\mathcal{F}$ of complex algebraic curves of genus 6 equipped with an automorphism of order 5. This is a family with 2 parameters, and is fibred over a certain type of Del Pezzo surace. The period satisfies the hypergeometric differential equation for Appell's $F_1({3/5},{3/5},{2/5},{6/5})$ of two variables after a certain normalization of the variable parameter. This differential equation and the family $\mathcal{F}$ are studied by G. Shimura (1964), T. Terada (1983, 1985), P. Deligne - G.D. Mostow (1986) and T. Yamazaki- M. Yoshida(1984). Recently M. Yoshida presented a new approch using the concept of configration space. Based on their results we show the representation of the inverse of the period map in terms of Riemann theta constants. This is the first variant of the work of H. Shiga (1981) and K. Matsumoto (1989, 2000) to the co-compact case. | On the Family of Pentagonal curves of genus 6 and associated modular
forms on the Ball | 12,370 |
For a compact Riemann surface $X$ of genus $g > 1$, $\Hom(\pi_1(X), PU(p,q))/PU(p,q)$ is the moduli space of flat $PU(p,q)$-connections on $X$. There are two invariants, the Chern class $c$ and the Toledo invariant $\tau$ associated with each element in the moduli. The Toledo invariant is bounded in the range $-2min(p,q)(g-1) \le \tau \le 2min(p,q)(g-1)$. This paper shows that the component, associated with a fixed $\tau > 2(max(p,q)-1)(g-1)$ (resp. $\tau < -2(max(p,q)-1)(g-1)$) and a fixed Chern class $c$, is connected (The restriction on $\tau$ implies $p=q$). | The Moduli of Flat PU(p,p)-Structures with Large Toledo Invariants | 12,371 |
A component of the moduli space M_g(Y,b) of stable maps from genus g curves to a variety Y is said to be regular if it is generically smooth and of the expected dimension provided by deformation theory. In this note we prove existence of regular components of M_g(Y,b) when Y is a product of projective spaces. This result can also be seen as a general position theorem for Brill-Noether loci in M_g. | Regular components of moduli spaces of stable maps | 12,372 |
Suppose that $\Phi:X\to Y$ is a morphism from a 3 fold to a surface (over an algebraically closed field of characteristic zero). We prove that there exist sequences of blowups of nonsingular subvarieties $X_1\to X$ and $Y_1\to Y$ such that the morphism $\Phi_1:X_1\to Y_1$ is a ``Monomial Morphism''. As a corollary, we show that we can make $\Phi_1$ a toroidal morphism. | Monomialization of Morphisms From 3 Folds to Surfaces | 12,373 |
In this paper we prove the topological uniqueness of maximal arrangements of a real plane algebraic curve with respect to three lines. More generally, we prove the topological uniqueness of a maximally arranged algebraic curve on a real toric surface. We use the moment map as a tool for studying the topology of real algebraic curves and their complexifications. | Real algebraic curves, the moment map and amoebas | 12,374 |
We show that the moduli space of rational elliptic surfaces admitting a section is locally a complex hyperbolic variety of dimension eight. We compare its Satake-Baily-Borel compactification with a compactification obtained by means of geometric invariant theory, considered by Miranda. | The moduli space of rational elliptic surfaces | 12,375 |
Let k be a p-adic field. Some time ago, D. Harbater [9] proved that any finite group G may be realized as a regular Galois group over the rational function field in one variable k(t), namely there exists a finite field extension $F/k(t)$, Galois with group G, such that F is a regular extension of k (i.e. k is algebraically closed in F). Moreover, one may arrange that a given k-place of k(t) be totally split in F. Harbater proved this theorem for k an arbitrary complete valued field. Rather formal arguments ([10, \S 4.5]; \S2 hereafter) then imply that the theorem holds over any `large' field k. This in turn is a special case of a result of Pop [15], hence will be referred to as the Harbater/Pop theorem. We refer to [10], [16], [6] for precise references to the literature (work of D\`ebes, Deschamps, Fried, Haran, Harbater, Jarden, Liu, Pop, Serre, and V\"olklein). Most proofs (see [10], [19, 8.4.4, p.~93] and Liu's contribution to [16]; see however [15]) first use direct arguments to establish the theorem when G is a cyclic group (here the nature of the ground field is irrelevant), then proceed by patching, using either formal or rigid geometry, together with GAGA theorems. In the present paper, where I take the case of algebraically closed fields for granted, I show how a technique recently developed by Koll\'ar [12] may be used to give a quite different proof of the Harbater/Pop theorem, when the `large' field k has characteristic zero. This proof actually gives more than the original result (see comment after statement of Theorem 1). | Rational connectedness and Galois covers of the projective line | 12,376 |
This paper forms the major portion of a talk given at the International Colloquium on Arithmetic, Algebra and Geometry at TIFR, Mumbai in Jan 2000. We look at the problem of detecting cycles with trivial Abel-Jacobi invariant. M. Green proposed a Hodge-theoretic method to which C. Voisin found a counter-example. We present an easier example. We also propose another possible invariant to detect these classes using Hodge Theory. Similar methods have been proposed earlier by M. Asakura and M. Saito. | Higher Abel-Jacobi Maps | 12,377 |
The article investigates the following question: given a projective variety X acted on by a connected and reductive group G, which is the relationship between the Gromov-Witten invariants of X and those of X//G? In this study we shall also try to give an algebraic approach to the so called Hamiltonian Gromov-Witten invariants which have appeared rather recenly in the literature. | GW Invariants and Invariant Quotients | 12,378 |
We study one parameter deformations of a pair consisting of an analytic singular space $X_0$ and a function $f_0$ on it, in case this defines an isolated singularity. We prove, under general conditions, a bouquet decomposition of the Milnor fibre when the isolated singularity splits in the deformation and the invariance of the Milnor fibration if there is no splitting. | Splitting of Singularities | 12,379 |
Let $k$ be an integer such that $1\leq k\leq n-5$, and $X_{2n-2-k}\subset \mathbf P^n$ a general projective hypersurface of degree $d=2n-2-k$. In this paper we prove that the only $k$-dimensional subvariety $Y$ of $X_{2n-2-k}$ having geometric genus zero is the one covered by the lines. As an immediate corollary we obtain that, for $n>5$, the general $X_{2n-3}\subset \mathbf P^n$, contains no rational curves of degree $\delta >1$. | Rational curves on general projective hypersurfaces | 12,380 |
For a finite Abelian subgroup A of SL(3,C), Ito and Nakajima proved that the tautological bundles on the A-Hilbert scheme Y = A-Hilb(C^3) form a basis of the K-theory of Y. We establish the relations between these bundles in the Picard group of Y and hence, following a recipe introduced by Reid, construct an explicit basis of the integral cohomology of Y in one-to-one correspondence with the irreducible representations of A. | An explicit construction of the McKay correspondence for A-Hilb C^3 | 12,381 |
This note briefly reviews the {\it Mirror Principle} as developed in the series of papers \LLYI\LLYII\LLYIII\LLYIV\LCHY. We illustrate this theory with a few new examples. One of them gives an intriguing connection to a problem of counting holomorphic disks and annuli. This note has been submitted for the proceedings of the Workshop on Strings, Duality and Geometry the C.R.M. in Montreal of March 2000. | A Survey of Mirror Principle | 12,382 |
The (small) quantum cohomology ring of a flag manifold F encodes enumerative geometry of rational curves on F. We give a proof of the presentation of the ring and of a quantum Giambelli formula, which is more direct and geometric than the previously known proof. Furthermore, this proof gives a relationship between quantum Schubert polynomials and universal Schubert polynomials, which arise in a degeneracy locus formula of Fulton. | Quantum cohomology of flag manifolds | 12,383 |
Let $X$ be a smooth $n$-dimensional projective variety over an algebraically closed field $k$ such that $K_X$ is not nef. We give a characterization of non nef extremal rays of $X$ of maximal length (i.e of length $n-1$); in the case of $\ch(k) = 0$ we also characterize non nef rays of length $n-2$. | Special rays in the Mori cone of a projective variety | 12,384 |
The notion of a holomorphically symplectic manifold can be generalized to the singular one. This paper studies the birational contraction maps between symplectic varieties, and then describes the deformation of a symplectic variety which has a symplectic resolution. Some examples are also included. | Deformation theory of singular symplectic n-folds | 12,385 |
This paper deals with symplectic varieties which do not have symplectic resolutions. Some moduli spaces of semi-stable torsion-free sheaves on a K3 surface, and symplectic V-manifolds are such varieties. We shall prove local Torelli theorem for symplectic varieties. Some results on symplectic singularities are also included. | Extension of 2-forms and symplectic varieties | 12,386 |
A stratification of a singular set, e.g. an algebraic or analytic variety, is, roughly, a partition of it into manifolds so that these manifolds fit together "regularly". A classical theorem of Whitney says that any complex analytic set has a stratification. This result was extended by Lojasiewicz to real (semi)analytic sets. In this paper we present a short geometric proof of existence of stratifications based on Thom's transversality theorem and Milnor's curve selection lemma and not relying on difficult results of Lojasiewicz. | A geometric proof of the existence of Whitney stratifications | 12,387 |
In this paper we compute upper bounds for the number of ordinary triple points on a hypersurface in $P^3$ and give a complete classification for degree six (degree four or less is trivial, and five is elementary). But the real purpose is to point out the intricate geometry of examples with many triple points, and how it fits with the general classification of surfaces. | Surfaces with triple points | 12,388 |
We show that the fundamental group of the complement of any irreducible tame torus sextics in $\bf P^2$ is isomorphic to $\bf Z_2*\bf Z_3$ except one class. The exceptional class has the configuration of the singularities $\{C_{3,9},3A_2\}$ and the fundamental group is bigger than $\bf Z_2*\bf Z_3$. In fact, the Alexander polynomial is given by $(t^2-t+1)^2$. For the proof, we first reduce the assertion to maximal curves and then we compute the fundamental groups for maximal tame torus curves. | Fundamental group of sextics of torus type | 12,389 |
In this paper, we address the following two general problems: given two algebraic varieties in ${\bf C}^n$, find out whether or not they are (1) isomorphic; (2) equivalent under an automorphism of ${\bf C}^n$. Although a complete solution of either of these problems is out of the question at this time, we give here some handy and useful invariants of isomorphic as well as of equivalent varieties. Furthermore, and more importantly, we give a universal procedure for obtaining all possible algebraic varieties isomorphic to a given one, and use it to construct numerous examples of isomorphic, but inequivalent algebraic varieties in ${\bf C}^n$. Among other things, we establish the following interesting fact: for isomorphic hypersurfaces $\{p(x_1,...,x_n)=0\}$ and $\{q(x_1,...,x_n)=0\}$, the number of zeros of $grad(p)$ might be different from that of $grad(q)$. | Embeddings of hypersurfaces in affine spaces | 12,390 |
Let $X$ be a projective variety, homogeneous under a linear algebraic group. We show that the diagonal of $X$ belongs to a unique irreducible component $H_X$ of the Hilbert scheme of $X\times X$. Moreover, $H_X$ is isomorphic to the ``wonderful completion'' of the connected automorphism group of $X$; in particular, $H_X$ is non-singular. We describe explicitly the degenerations of the diagonal in $X\times X$, that is, the points of $H_X$; these subschemes of $X\times X$ are reduced and Cohen-Macaulay. | Group completions via Hilbert schemes | 12,391 |
Given an algorithm of resolution of singularities satisfying certain conditions (``good algorithms''), natural notions of simultaneous algorithmic resolution, or equiresolution, for families of embedded schemes (parametrized by a reduced scheme $T$) are proposed. It is proved that these conditions are equivalent. Something similar is done for families of sheaves of ideals, here the goal is algorithmic simultaneous principalization. A consequence is that given a family of embedded schemes over a reduced $T$, this parameter scheme can be naturally expressed as a disjoint union of locally closed sets $T_{j}$, such that the induced family on each part $T_{j}$ is equisolvable. In particular, this can be applied to the Hilbert scheme of a smooth projective variety; in fact, our result shows that, in characteristic zero, the underlying topological space of any Hilbert scheme parametrizing embedded schemes can be naturally stratified in equiresolvable families. | On Algorithmic Equiresolution and Stratification of Hilbert Schemes | 12,392 |
By means of a Fourier-Mukai transform we embed moduli spaces of stable bundles on an algebraic curve C as isotropic subvarieties of moduli spaces of mu-stable bundles on the Jacobian variety J(C). When g(C)=2 this provides new examples of special Lagrangian submanifolds. | Complex Lagrangian embeddings of moduli spaces of vector bundles | 12,393 |
We prove that the higher direct images of the dualizing sheaf of a Lagrangian fibration between smooth projective manifolds are isomorphic to the cotangent bundles of base space. As a corollary, we obtain that every Hodge number of the base space of a fibre space of an irreducible symplectic manifold is the same to that of a projective space if the base space is smooth. | Higher direct images of Lagrangian fibrations | 12,394 |
We use pluriharmonic maps to study representations of fundamental groups of algebraic manifolds. This approach is functorial in the sense that the restriction of such a map to a fiber of a fibration remains pluriharmonic, and on this basis, we can investigate the relation between the properties of the original representation and what can be inferred by restricting it to the fibers of a fibration. | Representations of fundamental groups of algebraic manifolds and their
restrictions to fibers of a fibration | 12,395 |
We give a characteristic-free proof that general codimension-1 Schubert varieties meet transversally in a Grassmannian and in some related varieties. Thus the corresponding intersection numbers computed in the Chow (and quantum Chow) rings of these varieties are enumerative in all characteristics. We show that known transversality results do not apply to these enumerative problems, emphasizing the need for additional theoretical work on transversality. The method of proof also strengthens some results in real enumerative geometry. | Elementary Transversality in the Schubert calculus in any Characteristic | 12,396 |
We consider the variety of $(p+1)$-tuples of matrices $M_j$ from given conjugacy classes from $GL(n,{\bf C})$ such that $M_1... M_{p+1}=I$. This variety is connected with the Deligne-Simpson problem and the matrices $M_j$ are interpreted as monodromy operators of regular systems on Riemann's sphere. We consider among others cases when the dimension of the variety is higher than the expected one due to the presence of $(p+1)$-tuples with non-trivial centralizers. | Some examples related to the Deligne-Simpson problem | 12,397 |
The Mellin transform of fibre integral is calculated for certain isolated singularities of quasihomogeneous complete intersections (especially the unimodal singualrities of the list by Giusti and Wall). We show the property of symmetry between spectra of Gauss-Manin and shed light on the lattice structure of poles of the Mellin transform by means of topological data of the singularity. As an application, Hodge numbers of the Milnor fibre are expressed by counting the spectra of Gauss-Manin. | Transformée de Mellin des intégrales-fibres associées aux
singularités isolées d'intersection comlète quasihomogène | 12,398 |
We study algebraic fiber spaces $f:X \longrightarrow Y$ where $Y$ is of maximal Albanese dimension. In particular we give an effective version a theorem of Kawamata: If $P_m(X)=1$ for some $m \ge 2$, then the Albanese map of $X$ is surjective. Combining this with \cite{CH} it follows that $X$ is birational to an abelian variety if and only if $P_2(X)=1$ and $q(X)= \text{\rm dim} (X)$. | On algebraic fiber spaces over varieties of maximal Albanese dimension | 12,399 |
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