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Theorem 2.6 Let \( X \in {\mathrm{{VII}}}_{ > 0}^{\min } \) with \( 1 \leq {b}_{2}\left( X\right) \leq 3 \), and let \( g \) be a Gauduchon metric on \( X \) with \( {\deg }_{g}\left( {\mathcal{K}}_{X}\right) < 0 \) . Suppose that \( \mathcal{A} \) is stable, that it cannot be written as an extension of the form (1) wi... | By the main result of [30], the existence of such a subspace leads to a contradiction. This results holds in full generality (for surfaces with arbitrary positive second Betti number). Taking into account Corollaries 2.2 - 2.4, we obtain the following result, which proves Conjecture 3 for class VII surfaces with small ... | No |
Proposition 1.1.2 Let \( f : \overline{{B}_{\varepsilon }\left( w\right) } \rightarrow \mathbb{C} \) be a continuous function such that \( f \) is holomorphic with respect to every single component \( {z}_{i} \) in any point of \( {B}_{\varepsilon }\left( w\right) \) . Then for any \( z \in {B}_{\varepsilon }\left( w\r... | Proof. Repeated application of the Cauchy integral formula in one variable yields\n\n\[ f\left( z\right) = \frac{1}{{\left( 2\pi i\right) }^{n}}{\int }_{\left| {{\xi }_{1} - {w}_{1}}\right| = {\varepsilon }_{1}}\ldots {\int }_{\left| {{\xi }_{n} - {w}_{n}}\right| = {\varepsilon }_{n}}\frac{f\left( {{\xi }_{1},\ldots ,{... | Yes |
Lemma 1.1.3 Let \( U \subset {\mathbb{C}}^{n} \) be an open subset and let \( V \subset \mathbb{C} \) be an open neighbourhood of the boundary of \( {B}_{\varepsilon }\left( 0\right) \subset \mathbb{C} \) . Assume that \( f : V \times U \rightarrow \mathbb{C} \) is a holomorphic function. Then\n\n\[ g\left( z\right) \m... | Proof. Let \( z \in U \) . If \( \left| \xi \right| = \varepsilon \) then there exists a polydisc \( {B}_{\delta \left( \xi \right) }\left( \xi \right) \times {B}_{{\delta }^{\prime }\left( \xi \right) }\left( z\right) \subset \) \( V \times U \) on which \( f \) has a power series expansion.\n\nSince \( \partial {B}_{... | Yes |
Proposition 1.1.4 (Hartogs’ theorem) Suppose \( \varepsilon = \left( {{\varepsilon }_{1},\ldots ,{\varepsilon }_{n}}\right) \) and \( {\varepsilon }^{\prime } = \) \( \left( {{\varepsilon }_{1}^{\prime },\ldots ,{\varepsilon }_{n}^{\prime }}\right) \) are given such that for all \( i \) one has \( {\varepsilon }_{i}^{\... | Proof. We may assume that \( \varepsilon = \left( {1,\ldots ,1}\right) \) . Moreover, there exists \( \delta > 0 \) such that the open subset \( V \mathrel{\text{:=}} \left\{ {z\left| {1 - \delta < }\right| {z}_{1}\left| { < 1,}\right| {z}_{i \neq 1} \mid < 1}\right\} \cup \{ z \mid 1 - \delta < \) \( \left. {\left| {z... | Yes |
Proposition 1.1.7 (Riemann extension theorem) Let \( f \) be a holomorphic function on an open subset \( U \subset {\mathbb{C}}^{n} \) . If \( g : U \smallsetminus Z\left( f\right) \rightarrow \mathbb{C} \) is holomorphic and locally bounded near \( Z\left( f\right) \), then \( g \) can uniquely be extended to a holomo... | Proof. Before launching into the proof let us consider the following special case: \( n = 2 \) and \( f\left( z\right) = {z}_{1} \) . Then \( {g}_{{z}_{2}}\left( {z}_{1}\right) \mathrel{\text{:=}} g\left( {{z}_{1},{z}_{2}}\right) \) is a bounded holomorphic function on a punctured disc in the complex plane. Thus, by th... | Yes |
Proposition 1.1.13 Let \( f : U \rightarrow V \) be a bijective holomorphic map between two open subsets \( U, V \subset {\mathbb{C}}^{n} \) . Then for all \( z \in U \) one has \( \det J\left( f\right) \left( z\right) \neq 0 \) . In particular, \( f \) is biholomorphic. | Proof. The proof proceeds by induction. For \( n = 1 \) this is standard, but for completeness sake we recall the argument. Suppose \( {f}^{\prime } \) has a zero. After a suitable coordinate change, we can assume \( f\left( 0\right) = {f}^{\prime }\left( 0\right) = 0 \) . Then the power series expansion of \( f \) has... | No |
Proposition 1.1.15 The local ring \( {\mathcal{O}}_{{\mathbb{C}}^{n},0} \) is a UFD. | Proof. We prove the assumption by induction on \( n \) . For \( n = 0 \) the ring \( {\mathcal{O}}_{{\mathbb{C}}^{n},0} = \) \( \mathbb{C} \) is a field and thus a UFD. Suppose that \( {\mathcal{O}}_{{\mathbb{C}}^{n - 1},0} \) is a UFD. If \( f \in {\mathcal{O}}_{{\mathbb{C}}^{n},0} \) we can choose coordinates such th... | Yes |
Proposition 1.1.17 (Weierstrass division theorem) Let \( f \in {\mathcal{O}}_{{\mathbb{C}}^{n},0} \) and let \( g \in {\mathcal{O}}_{{\mathbb{C}}^{n - 1},0}\left\lbrack {z}_{1}\right\rbrack \) be a Weierstrass polynomial of degree \( d \) . Then there exist \( r \in {\mathcal{O}}_{{\mathbb{C}}^{n - 1},0}\left\lbrack {z... | Proof. The uniqueness is easy. Assume \( f = g \cdot {h}_{1} + {r}_{1} = g \cdot {h}_{2} + {r}_{2} \) . Then \( {r}_{1} - {r}_{2} = g \cdot \left( {{h}_{2} - {h}_{1}}\right) \) . For \( w = \left( {{z}_{2},\ldots ,{z}_{n}}\right) \) we consider the function \( {g}_{w}\left( {z}_{1}\right) \mathrel{\text{:=}} \) \( g\le... | Yes |
Proposition 1.1.18 The local UFD \( {\mathcal{O}}_{{\mathbb{C}}^{n},0} \) is noetherian. | Proof. We have to show that any ideal in \( {\mathcal{O}}_{{\mathbb{C}}^{n},0} \) is finitely generated. We use again induction on \( n \) . The case \( n = 0 \) is trivial, as any field is noetherian.\n\nNext, we assume that \( {\mathcal{O}}_{{\mathbb{C}}^{n - 1},0} \) is noetherian. Then, also the polynomial ring \( ... | Yes |
Corollary 1.1.19 Let \( g \in {\mathcal{O}}_{{\mathbb{C}}^{n},0} \) be an irreducible function. If \( f \in {\mathcal{O}}_{{\mathbb{C}}^{n},0} \) vanishes on \( Z\left( g\right) \), then \( g \) divides \( f \) . | Proof. By the WPT (Proposition 1.1.6) we may assume that \( g \in {\mathcal{O}}_{{\mathbb{C}}^{n - 1},0}\left\lbrack {z}_{1}\right\rbrack \) is a Weierstrass polynomial of degree \( d \) . By the Weierstrass division theorem (Proposition 1.1.17) one finds \( h \in {\mathcal{O}}_{{\mathbb{C}}^{n},0} \) and \( r \in {\ma... | No |
Lemma 1.1.25 For any germ \( X \subset {\mathbb{C}}^{n} \) the set \( I\left( X\right) \subset {\mathcal{O}}_{{\mathbb{C}}^{n},0} \) is an ideal. If \( \left( A\right) \subset {\mathcal{O}}_{{\mathbb{C}}^{n},0} \) denotes the ideal generated by a subset \( A \subset {\mathcal{O}}_{{\mathbb{C}}^{n},0} \), then \( Z\left... | Proof. All assertions are easily verified. Except, perhaps, the last one. Here one has to use the fact that \( {\mathcal{O}}_{{\mathbb{C}}^{n},0} \) is noetherian, i.e. any ideal is generated by finitely many elements. | No |
Lemma 1.1.26 If \( {X}_{1} \subset {X}_{2} \), then \( I\left( {X}_{2}\right) \subset I\left( {X}_{1}\right) \) . If \( {I}_{1} \subset {I}_{2} \), then \( Z\left( {I}_{2}\right) \subset Z\left( {I}_{1}\right) \) . For any analytic germ \( X \) one has \( Z\left( {I\left( X\right) }\right) = X \) . For any ideal \( I \... | Proof. The first two assertions are obvious. Clearly, \( X \subset Z\left( {I\left( X\right) }\right) \) . On the other hand, there exist elements \( {f}_{1},\ldots ,{f}_{k} \in {\mathcal{O}}_{{\mathbb{C}}^{n},0} \) with \( X = Z\left( {{f}_{1},\ldots ,{f}_{k}}\right) \) . Then, \( {f}_{1},\ldots ,{f}_{k} \in I\left( X... | Yes |
Lemma 1.1.28 An analytic germ \( X \) is irreducible if and only if \( I\left( X\right) \subset {\mathcal{O}}_{{\mathbb{C}}^{n},0} \) is a prime ideal. | Proof. If \( X \) is irreducible and \( {f}_{1} \cdot {f}_{2} \in I\left( X\right) \), then \( X = \left( {X \cap Z\left( {f}_{1}\right) }\right) \cup \left( {X \cap Z\left( {f}_{2}\right) }\right) \) is a union of analytic germs. Thus, \( X = X \cap Z\left( {f}_{i}\right) \) for \( i = 1 \) or \( i = 2 \) . Hence, at ... | Yes |
Proposition 1.1.35 Let \( f \in {\mathcal{O}}_{{\mathbb{C}}^{n},0} \) be irreducible. Then for sufficiently small \( \varepsilon \) and \( z \in {B}_{\varepsilon }\left( 0\right) \) the induced element \( f \in {\mathcal{O}}_{{\mathbb{C}}^{n}, z} \) is irreducible. | Proof. We may assume that \( f \in {\mathcal{O}}_{{\mathbb{C}}^{n - 1},0}\left\lbrack {z}_{1}\right\rbrack \) is a Weierstrass polynomial. Suppose that \( f \) as an element of \( {\mathcal{O}}_{{\mathbb{C}}^{n}, z} \) is reducible. Then \( f = {f}_{1} \cdot {f}_{2} \) with \( {f}_{i} \in {\mathcal{O}}_{{\mathbb{C}}^{n... | Yes |
Proposition 1.1.36 (Schwarz lemma) Let \( \varepsilon \mathrel{\text{:=}} \left( {\delta ,\ldots ,\delta }\right) \) and let \( f \) be a holomorphic function on an open neighbourhood of the closure of the polydisc \( \overline{{B}_{\varepsilon }\left( 0\right) } \) . Assume that \( f \) vanishes of order \( k \) at th... | Proof. Fix \( 0 \neq z \in {B}_{\delta }\left( 0\right) \) and define a holomorphic function \( {g}_{z} \) of one variable as follows: For \( w \leq \delta \) one sets\n\n\[ {g}_{z}\left( w\right) \mathrel{\text{:=}} {w}^{-k}f\left( {w \cdot \frac{z}{\left| z\right| }}\right) . \]\n\nThen \( \left| {{g}_{z}\left( w\rig... | Yes |
Lemma 1.2.2 If \( I \) is an almost complex structure on a real vector space \( V \) , then \( V \) admits in a natural way the structure of a complex vector space. | Proof. The \( \mathbb{C} \) -module structure on \( V \) is defined by \( \left( {a + {ib}}\right) \cdot v = a \cdot v + b \cdot I\left( v\right) \) , where \( a, b \in \mathbb{R} \) . The \( \mathbb{R} \) -linearity of \( I \) and the assumption \( {I}^{2} = - \mathrm{{id}} \) yield \( \left( {\left( {a + {ib}}\right)... | Yes |
Corollary 1.2.3 Any almost complex structure on \( V \) induces a natural orientation on \( V \) . | Proof. Using the lemma, the assertion reduces to the statement that the real vector space \( {\mathbb{C}}^{n} \) admits a natural orientation. We may assume \( n = 1 \) and use the orientation given by the basis \( \left( {1, i}\right) \) . The orientation is well-defined, as it does not change under \( \mathbb{C} \) -... | Yes |
Lemma 1.2.5 Let \( V \) be a real vector space endowed with an almost complex structure I. Then\n\n\[ \n{V}_{\mathbb{C}} = {V}^{1,0} \oplus {V}^{0,1} \n\]\n\nComplex conjugation on \( {V}_{\mathbb{C}} \) induces an \( \mathbb{R} \) -linear isomorphism \( {V}^{1,0} \cong {V}^{0,1} \) . | Proof. Since \( {V}^{1,0} \cap {V}^{0,1} = 0 \), the canonical map\n\n\[ \n{V}^{1,0} \oplus {V}^{0,1} \rightarrow {V}_{\mathbb{C}} \n\]\n\nis injective. The first assertion follows from the existence of the inverse map\n\n\[ \nv \mapsto \frac{1}{2}\left( {v - {iI}\left( v\right) }\right) \oplus \frac{1}{2}\left( {v + {... | Yes |
Lemma 1.2.6 Let \( V \) be a real vector space endowed with an almost complex structure \( I \) . Then the dual space \( {V}^{ * } = {\operatorname{Hom}}_{\mathbb{R}}\left( {V,\mathbb{R}}\right) \) has a natural almost complex structure given by \( I\left( f\right) \left( v\right) = f\left( {I\left( v\right) }\right) \... | \[ {\left( {V}^{ * }\right) }^{1,0} = \left\{ {f \in {\operatorname{Hom}}_{\mathbb{R}}\left( {V,\mathbb{C}}\right) \mid f\left( {I\left( v\right) }\right) = {if}\left( v\right) }\right\} = {\left( {V}^{1,0}\right) }^{ * } \] \[ {\left( {V}^{ * }\right) }^{0,1} = \left\{ {f \in {\operatorname{Hom}}_{\mathbb{R}}\left( {V... | Yes |
Proposition 1.2.8 For a real vector space \( V \) endowed with an almost complex structure I one has:\n\ni) \( \mathop{\bigwedge }\limits^{{p, q}}V \) is in a canonical way a subspace of \( \mathop{\bigwedge }\limits^{{p + q}}{V}_{\mathbb{C}} \) .\n\nii) \( \mathop{\bigwedge }\limits^{k}{V}_{\mathbb{C}} = \mathop{\bigo... | Proof. Let \( {v}_{1},\ldots ,{v}_{n} \in \mathop{\bigwedge }\limits^{{1,0}}V = {V}^{1,0} \) and \( {w}_{1},\ldots ,{w}_{n} \in \mathop{\bigwedge }\limits^{{0,1}}V = {V}^{0,1} \) be \( \mathbb{C} \) -basis. Then \( {v}_{{J}_{1}} \otimes {w}_{{J}_{2}} \in \mathop{\bigwedge }\limits^{{p, q}}V \) with \( {J}_{1} = \left\{... | Yes |
Lemma 1.2.9 For any \( m \leq {\dim }_{\mathbb{C}}{V}^{1,0} \) one has\n\n\[{\left( -2i\right) }^{m}\left( {{z}_{1} \land {\bar{z}}_{1}}\right) \land \ldots \land \left( {{z}_{m} \land {\bar{z}}_{m}}\right) = \left( {{x}_{1} \land {y}_{1}}\right) \land \ldots \land \left( {{x}_{m} \land {y}_{m}}\right) .\]\n\nFor \( m ... | Proof. This is a straightforward calculation using induction on \( m \) . | No |
Lemma 1.2.14 Let \( \\left( {V,\\langle \\rangle ,\\rangle }\\right) \) be an euclidian vector space endowed with a compatible almost complex structure. Then, its fundamental form \( \\omega \) is real and of type \( \\left( {1,1}\\right) \), i.e. \( \\omega \\in {\\bigwedge }^{2}{V}^{ * } \\cap {\\bigwedge }^{1,1}{V}^... | Proof. Since\n\n\[ \n\\langle v, I\\left( w\\right) \\rangle ) = \\langle I\\left( v\\right), I\\left( {I\\left( w\\right) }\\right) \\rangle = - \\langle I\\left( v\\right), w\\rangle = - \\langle w, I\\left( v\\right) \\rangle \n\] \n\nfor all \( v, w \\in V \), the form \( \\omega \) is alternating, i.e. \( \\omega ... | Yes |
Lemma 1.2.15 Let \( \left( {V,\langle \rangle ,\rangle }\right) \) be an euclidian vector space endowed with a compatible complex structure. The form \( \left( {\cdot , \cdot }\right) \mathrel{\text{:=}} \langle \cdot , \cdot \rangle - i \cdot \omega \) is a positive hermitian form on \( \left( {V, I}\right) \) . | Proof. The form \( \left( { : , : }\right) \) is clearly \( \mathbb{R} \) -linear and \( \left( {v, v}\right) = \langle v, v\rangle > 0 \) for \( 0 \neq v \in V \) . Moreover, \( \left( {v, w}\right) = \overline{\left( w, v\right) } \) and\n\n\[ \left( {I\left( v\right), w}\right) = \langle I\left( v\right), w\rangle -... | Yes |
Lemma 1.2.16 If \( \\left( {V,\\langle \\;,\\;\\rangle }\\right) \) is an euclidian vector space with a compatible almost complex structure \( I \) . Then \( {V}_{\\mathbb{C}} = {V}^{1,0} \\oplus {V}^{0,1} \) is an orthogonal decomposition with respect to the hermitian product \( \\langle \) , \( {\\rangle }_{\\mathbb{... | Proof. Let \( v - {iI}\\left( v\\right) \\in {V}^{1,0} \) and \( w + {iI}\\left( w\\right) \\in {V}^{0,1} \) with \( v, w \\in V \) . Then an easy calculation shows \( \\langle v - {iI}\\left( v\\right), w + {iI}\\left( w\\right) {\\rangle }_{\\mathbb{C}} = 0 \) . | Yes |
Lemma 1.2.17 Let \( \left( {V,\langle \;,\;\rangle }\right) \) be an euclidian vector space with a compatible almost complex structure \( I \) . Under the canonical isomorphism \( \left( {V, I}\right) \cong \left( {{V}^{1,0}, i}\right) \) one has \( \frac{1}{2}\left( {\text{,}\omega }\right) = \langle \) , \( {\left. {... | Proof. The natural isomorphism was given by \( v \mapsto \frac{1}{2}\left( {v - {iI}\left( v\right) }\right) \) . Now use the definitions of \( \left( {\text{,}\beta }\right) {toconclude} \n\n\[ \n{\left\langle \left( v - iI\left( v\right) \right) ,\left( {v}^{\prime } - iI\left( {v}^{\prime }\right) \right) \right\ran... | Yes |
Lemma 1.2.23 The dual Lefschetz operator \( A \) is of degree -2, i.e. \( A\left( {\mathop{\bigwedge }\limits^{k}{V}^{ * }}\right) \subset \) \( \mathop{\bigwedge }\limits^{{k - 2}}{V}^{ * } \) . Moreover, one has \( \Lambda = { * }^{-1} \circ L \circ * \) . | Proof. The first assertion follows from the fact that \( L \) is of degree two and that \( {\bigwedge }^{ * }{V}^{ * } = \bigoplus {\bigwedge }^{k}{V}^{ * } \) is orthogonal.\n\nBy definition of the Hodge \( * \) -operator one has \( \langle \alpha ,{L\beta }\rangle \cdot \operatorname{vol} = \langle {L\beta },\alpha \... | Yes |
Lemma 1.2.24 Let \( \langle \) , \( {\rangle }_{\mathbb{C}},\Lambda \), and \( * \) be as above. Then\ni) The decomposition \( \mathop{\bigwedge }\limits^{k}{V}_{\mathbb{C}}^{ * } = \bigoplus \mathop{\bigwedge }\limits^{{p, q}}{V}^{ * } \) is orthogonal with respect to \( \langle \) , \( {\rangle }_{\mathbb{C}} \).\nii... | Proof. The first assertion follows directly from Lemma 1.2.16. The third assertion follows from the first and the fact that \( \Lambda \) is the formal adjoint of \( L \) with respect to \( \langle \) , \( {\rangle }_{\mathbb{C}} \). For the second assertion use \( \alpha \land * \bar{\beta } = \langle \alpha ,\beta {\... | Yes |
Proposition 1.2.26 Let \( \left( {V,\langle \;,\;\rangle }\right) \) be an euclidian vector space endowed with a compatible almost complex structure I. Consider the following linear operators on \( {\bigwedge }^{ * }{V}^{ * } \) : The associated Lefschetz operator \( L \), its dual \( \Lambda \), and the counting opera... | Proof. Let \( \alpha \in \mathop{\bigwedge }\limits^{k}{V}^{ * } \) . Then \( \left\lbrack {H, L}\right\rbrack \left( \alpha \right) = \left( {k + 2 - n}\right) \left( {\omega \land \alpha }\right) - \omega \land \left( {\left( {k - n}\right) \alpha }\right) = \) \( {2\omega } \land \alpha \) . Analogously, \( \left\lb... | Yes |
Corollary 1.2.27 Let \( \left( {V,\langle \rangle ,\rangle, I}\right) \) be an euclidian vector space with a compatible almost complex structure. The action of \( L,\Lambda \), and \( H \) defines a natural \( \mathfrak{{sl}}\left( 2\right) \) -representation on \( \mathop{\bigwedge }\limits^{ * }{V}^{ * } \) . | Proof. Recall, that \( \mathfrak{{sl}}\left( 2\right) \) is the three-dimensional (over \( \mathbb{C} \) or over \( \mathbb{R} \) ) Lie algebra of all \( 2 \times 2 \) -matrices of trace zero. A basis is given by \( X = \left( \begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right), Y = \left( \begin{array}{ll} 0 & 0 \\ 1 ... | Yes |
Corollary 1.2.28 \( \left\lbrack {{L}^{i},\Lambda }\right\rbrack \left( \alpha \right) = i\left( {k - n + i - 1}\right) {L}^{i - 1}\left( \alpha \right) \) for all \( \alpha \in \mathop{\bigwedge }\limits^{k}{V}^{ * } \) . | Proof. This is easily seen by induction on \( i \) as follows:\n\n\[ \left\lbrack {{L}^{i},\Lambda }\right\rbrack \left( \alpha \right) = {L}^{i}{\Lambda \alpha } - \Lambda {L}^{i}\alpha \]\n\n\[ = L\left( {{L}^{i - 1}{\Lambda \alpha } - \Lambda {L}^{i - 1}\alpha }\right) + {L\Lambda }{L}^{i - 1}\alpha - {\Lambda L}{L}... | Yes |
Proposition 1.2.30 Let \( \\left( {V,\\langle \\rangle ,\\rangle, I}\\right) \) be an euclidian vector space of dimension \( {2n} \) with a compatible almost complex structure and let \( L \) and \( \\Lambda \) be the associated Lefschetz operators.\ni) There exists a direct sum decomposition of the form:\n\n\[ \n\\mat... | Proof. i) The easiest way to prove i) is to apply some small amount of representation theory. Since \( {\\bigwedge }^{ * }{V}_{\\mathbb{C}}^{ * } \) is a finite-dimensional \( \\mathfrak{{sl}}\\left( 2\\right) \) -representation, it is a direct sum of irreducible ones. Any finite-dimensional \( \\mathfrak{{sl}}\\left( ... | Yes |
Let \( j = k = 0 \) and \( \alpha = 1 \), then we obtain \( * 1 = \frac{1}{n!}{L}^{n}1 = \frac{{\omega }^{n}}{n!} \) . | Thus, vol \( = \frac{{\omega }^{n}}{n!} \) as was claimed before (Remark 1.2.22). | No |
Example 1.2.34 In particular, \( \mathop{\bigwedge }\limits^{0}{V}_{\mathbb{C}}^{ * } = {P}^{0,0} = {P}_{\mathbb{C}}^{0} = \mathbb{C},\mathop{\bigwedge }\limits^{1}{V}_{\mathbb{C}}^{ * } = {P}^{1,0} \oplus {P}^{0,1} \) , and | \[ \mathop{\bigwedge }\limits^{2}{V}_{\mathbb{C}}^{ * } = \mathop{\bigwedge }\limits^{{2,0}}{V}^{ * } \oplus \mathop{\bigwedge }\limits^{{1,1}}{V}^{ * } \oplus \mathop{\bigwedge }\limits^{{0,2}}{V}^{ * } \] \[ = {P}^{2,0} \oplus \left( {{P}^{1,1} \oplus \omega \mathbb{C}}\right) \oplus {P}^{0,2}. \] | Yes |
Corollary 1.2.36 (Hodge-Riemann bilinear relation) Let \( \left( {V,\langle \;,\;\rangle, I}\right) \;{be} \) an euclidian vector space endowed with a compatible almost complex structure. Then the associated Hodge-Riemann pairing \( Q \) satisfies:\n\n\[ Q\left( {\mathop{\bigwedge }\limits^{{p, q}}{V}^{ * },\mathop{\bi... | Proof. Only the second assertion needs a proof. By definition\n\n\[ Q\left( {\alpha ,\bar{\alpha }}\right) \cdot \mathrm{{vol}} = {\left( -1\right) }^{\frac{k\left( {k - 1}\right) }{2}}\alpha \land \bar{\alpha } \land {\omega }^{n - k} \]\n\n\[ = {\left( -1\right) }^{\frac{k\left( {k - 1}\right) }{2}}\alpha \land {L}^{... | Yes |
Example 1.2.37 Suppose \( n \geq 2 \) and consider the decomposition \( {\left( \mathop{\bigwedge }\limits^{{1,1}}{V}^{ * }\right) }_{\mathbb{R}} = \) \( \omega \mathbb{R} \oplus {P}_{\mathbb{R}}^{1,1} \), where ( \( {)}_{\mathbb{R}} \) denotes the intersection with \( \mathop{\bigwedge }\limits^{2}{V}^{ * } \) . Then,... | Moreover, \( Q \) is a positive definite symmetric bilinear form on \( \omega \mathbb{R} \) and a negative definite symmetric bilinear from on \( {P}_{\mathbb{R}}^{1,1} \) . This is what will lead to the Hodge index theorem in Section 3.3. | Yes |
Proposition 1.3.1 The complexified tangent bundle \( {T}_{\mathbb{C}}U \mathrel{\text{:=}} {TU} \otimes \mathbb{C} \) decomposes as a direct sum of complex vector bundles\n\n\[ {T}_{\mathbb{C}}U = {T}^{1,0}U \oplus {T}^{0,1}U \]\n\nsuch that the complex linear extension of \( I \) satisfies\n\n\[ {\left. I\right| }_{{T... | The vector bundles \( {T}^{1,0}U \) and \( {T}^{0,1}U \) are trivialized by the sections \( \frac{\partial }{\partial {z}_{i}} \mathrel{\text{:=}} \frac{1}{2}\left( {\frac{\partial }{\partial {x}_{i}} - i\frac{\partial }{\partial {y}_{i}}}\right) \) and \( \frac{\partial }{\partial {\bar{z}}_{i}} \mathrel{\text{:=}} \f... | Yes |
Lemma 1.3.6 For the differential operators \( \partial \) and \( \bar{\partial } \) one has:\n\ni) \( d = \partial + \bar{\partial } \) .\n\nii) \( {\partial }^{2} = {\bar{\partial }}^{2} = 0 \) and \( \partial \bar{\partial } = - \bar{\partial }\partial \) .\n\niii) They satisfy the Leibniz rule, i.e.\n\n\[ \n\partial... | Proof. i) follows from the local description of \( \partial \) and \( \bar{\partial } \) given above and ii) is deduced from \( {d}^{2} = 0 \) .\n\nTo see iii) we recall that the exterior differential satisfies\n\n\[ \nd\left( {\alpha \land \beta }\right) = d\left( \alpha \right) \land \beta + {\left( -1\right) }^{p + ... | Yes |
Let \( g \) be the constant standard metric such that\n\n\[ \frac{\partial }{\partial {x}_{1}},\ldots ,\frac{\partial }{\partial {x}_{n}},\frac{\partial }{\partial {y}_{1}},\ldots ,\frac{\partial }{\partial {y}_{n}} \]\n\nis an orthonormal basis for any \( {T}_{x}U \) . Clearly, complex structure and \( g \) are compat... | An arbitrary metric \( g \) on \( U \), if compatible with the almost complex structure, is uniquely determined by the matrix \( {h}_{ij}\left( z\right) \mathrel{\text{:=}} h\left( {\frac{\partial }{\partial {x}_{i}},\frac{\partial }{\partial {x}_{j}}}\right) \) . The fundamental form can then be written as\n\n\[ \omeg... | No |
Example 1.3.13 Any compatible metric on \( U \subset \mathbb{C} \) satisfies the above condition. | Clearly, the three-form \( {d\omega } \) vanishes for dimension reasons. | No |
Proposition 2.1.5 Let \( X \) be a compact connected complex manifold. Then \( \Gamma \left( {X,{\mathcal{O}}_{X}}\right) = \mathbb{C} \), i.e. any global holomorphic function on \( X \) is constant. | Proof. Since \( X \) is compact, any holomorphic function \( f : X \rightarrow \mathbb{C} \), which is in particular continuous, attains its maximum at some point \( x \in X \) . If \( \left( {{U}_{i},{\varphi }_{i}}\right) \) is a holomorphic chart with \( x \in {U}_{i} \), then \( f \circ {\varphi }_{i}^{-1} \) is lo... | Yes |
Proposition 2.2.6 The set \( \mathcal{O}\left( {-1}\right) \subset {\mathbb{P}}^{n} \times {\mathbb{C}}^{n + 1} \) that consists of all pairs \( \left( {\ell, z}\right) \in {\mathbb{P}}^{n} \times {\mathbb{C}}^{n + 1} \) with \( z \in \ell \) forms in a natural way a holomorphic line bundle over \( {\mathbb{P}}^{n} \) ... | Proof. The projection \( \pi : \mathcal{O}\left( {-1}\right) \rightarrow {\mathbb{P}}^{n} \) is given by projecting to the first factor. Let \( {\mathbb{P}}^{n} = \mathop{\bigcup }\limits_{{i = 0}}^{n}{U}_{i} \) be the standard open covering (see page 56). A canonical trivialization of \( \mathcal{O}\left( {-1}\right) ... | Yes |
Proposition 2.2.9 The tensor product and the dual endow the set of all isomorphism classes of holomorphic line bundles on a complex manifold \( X \) with the structure of an abelian group. This group is the Picard group \( \operatorname{Pic}\left( X\right) \) of \( X \). | Proof. By definition the product of two line bundles \( {L}_{1},{L}_{2} \) on \( X \) is the tensor product \( {L}_{1} \otimes {L}_{2} \) and the inverse of \( L \) is the dual \( {L}^{ * } \) . The only thing that needs a proof is that \( L \otimes {L}^{ * } \) is isomorphic to the trivial line bundle. This is best se... | No |
Corollary 2.2.10 There is a natural isomorphism \( \operatorname{Pic}\left( X\right) \cong {H}^{1}\left( {X,{\mathcal{O}}_{X}^{ * }}\right) \) . | Proof. The description of line bundles in terms of their cocycles provides us with an isomorphism \( \operatorname{Pic}\left( X\right) \cong {\check{H}}^{1}\left( {X,{\mathcal{O}}_{X}^{ * }}\right) \) . By general arguments, there always exists a natural homomorphism \( {\check{H}}^{p}\left( {X,\mathcal{F}}\right) \rig... | Yes |
Lemma 2.2.15 Let \( Y \subset X \) be a complex submanifold. Then there is a canonical injection \( {\mathcal{T}}_{Y} \subset {\mathcal{T}}_{X}{ \mid }_{Y} \) . | Proof. We apply ix) of Example 2.2.4 to our situation as follows. Let \( \left\{ \left( {{U}_{i},{\varphi }_{i}}\right) \right\} \) be a holomorphic atlas such that \( {\varphi }_{i}\left( {Y \cap {U}_{i}}\right) = \left\{ {z \mid {z}_{m + 1} = \ldots = {z}_{n} = }\right. \) \( 0\} \cap {\varphi }_{i}\left( {U}_{i}\rig... | Yes |
Proposition 2.2.17 (Adjunction formula) Let \( Y \) be a submanifold of a complex manifold \( X \) . Then the canonical bundle \( {K}_{Y} \) of \( Y \) is naturally isomorphic to the line bundle \( {\left. {K}_{X}\right| }_{Y} \otimes \det \left( {\mathcal{N}}_{Y/X}\right) \) . | Proof. Let \( \left\{ {\psi }_{ij}^{\prime }\right\} \) be a cocycle defining \( {\left. {\mathcal{T}}_{X}\right| }_{Y} \) as in the proof of Lemma 2.2.15. Then \( {\psi }_{ij}^{\prime } \) has the form \( \left( \begin{matrix} {\psi }_{ij} & * \\ 0 & {\phi }_{ij} \end{matrix}\right) \) . Here, \( \left\{ {\psi }_{ij}\... | Yes |
Proposition 2.2.19 Associating to a holomorphic vector bundle its sheaf of sections defines a canonical bijection between the set of holomorphic vector bundles of rank \( r \) and the set of locally free \( {\mathcal{O}}_{X} \) -modules of rank \( r \) . | Proof. First recall that a locally free \( {\mathcal{O}}_{X} \) -module of rank \( r \) is a sheaf \( \mathcal{F} \) of \( {\mathcal{O}}_{X} \) -modules on \( X \) which is locally isomorphic to \( {\mathcal{O}}_{X}^{\oplus r} \) . Clearly, the sheaf of sections of a holomorphic vector bundle \( \pi : E \rightarrow X \... | Yes |
Proposition 2.2.28 If \( X \) is a connected complex manifold, then there exists a natural inclusion \( {Q}_{0}\left( {R\left( X\right) }\right) \subset K\left( X\right) \) . Moreover, for \( R\left( X\right) \neq \mathbb{C} \) one has\n\n\[ \operatorname{kod}\left( X\right) = {\operatorname{trdeg}}_{\mathbb{C}}Q\left(... | Proof. We only need to show that \( {\operatorname{trdeg}}_{\mathbb{C}}Q\left( {R\left( X\right) }\right) - 1 = {\operatorname{trdeg}}_{\mathbb{C}}{Q}_{0}\left( {R\left( X\right) }\right) \) . Clearly, if \( {f}_{0},\ldots ,{f}_{k} \in Q\left( {R\left( X\right) }\right) \) are algebraically independent elements of degr... | Yes |
Proposition 2.3.9 There exists a natural isomorphism\n\n\[ \n{H}^{0}\left( {X,{\mathcal{K}}_{X}^{ * }/{\mathcal{O}}_{X}^{ * }}\right) \cong \operatorname{Div}\left( X\right) \n\] | Proof. The isomorphism is induced by associating to a meromorphic function its divisor. This is made precise as follows: An element \( f \in {H}^{0}\left( {X,{\mathcal{K}}_{X}^{ * }/{\mathcal{O}}_{X}^{ * }}\right) \) is given by non-trivial meromorphic functions \( {f}_{i} \in {\mathcal{K}}_{X}^{ * }\left( {U}_{i}\righ... | Yes |
Corollary 2.3.10 There exists a natural group homomorphism\n\n\\[ \n\\operatorname{Div}\\left( X\\right) \\rightarrow \\operatorname{Pic}\\left( X\\right) ,\\;D \\mapsto \\mathcal{O}\\left( D\\right) ,\n\\]\n\nwhere the definition of \\( \\mathcal{O}\\left( D\\right) \\) will be given in the proof. | Proof. If \\( D = \\sum {a}_{i}\\left\\lbrack {Y}_{i}\\right\\rbrack \\in \\operatorname{Div}\\left( X\\right) \\) corresponds to \\( f \\in {H}^{0}\\left( {X,{\\mathcal{K}}_{X}^{ * }/{\\mathcal{O}}_{X}^{ * }}\\right) \\), which in turn is given by functions \\( {f}_{i} \\in {\\mathcal{K}}_{X}^{ * }\\left( {U}_{i}\\rig... | Yes |
Proposition 2.3.12 Let \( f : X \rightarrow Y \) be a holomorphic map of connected complex manifolds and suppose that \( f \) is dominant, i.e. \( f\left( X\right) \) is dense in \( Y \) . Then the pull-back defines a group homomorphism\n\n\[ \n{f}^{ * } : \operatorname{Div}\left( Y\right) \rightarrow \operatorname{Div... | This description ii) of the pull-back shows that it is compatible with the group homomorphism \( \operatorname{Div}\left( X\right) \rightarrow \operatorname{Pic}\left( X\right) \) and the pull-back of line bundles (cf. Corollary 2.2.11). | No |
Lemma 2.3.14 A divisor \( D \in \operatorname{Div}\left( X\right) \) is principal if and only if \( \mathcal{O}\left( D\right) \cong \mathcal{O} \) . | Proof. If \( D \) is the principal divisor \( \left( f\right) \), then we may take as the corresponding section of \( {\mathcal{K}}_{X}^{ * }/{\mathcal{O}}_{X}^{ * } \) the image of \( f \in K\left( X\right) \) under the natural map \( K{\left( X\right) }^{ * } = \) \( {H}^{0}\left( {X,{\mathcal{K}}_{X}^{ * }}\right) \... | Yes |
Corollary 2.3.19 Non-trivial sections \( {s}_{1} \in {H}^{0}\left( {X,{L}_{1}}\right) \) and \( {s}_{2} \in {H}^{0}\left( {X,{L}_{2}}\right) \) define linearly equivalent divisors \( Z\left( {s}_{1}\right) \sim Z\left( {s}_{2}\right) \) if and only if \( {L}_{1} \cong {L}_{2} \) . | Proof. This is a consequence of \( \mathcal{O}\left( {Z\left( {s}_{i}\right) }\right) \cong {L}_{i} \) and Lemma 2.3.14. | No |
Corollary 2.3.20 The image of the natural map \( \operatorname{Div}\left( X\right) \rightarrow \operatorname{Pic}\left( X\right) \) is generated by those line bundles \( L \in \operatorname{Pic}\left( X\right) \) with \( {H}^{0}\left( {X, L}\right) \neq 0 \) . | Proof. Proposition 2.3.18 shows that any \( L \in \operatorname{Pic}\left( X\right) \) with \( {H}^{0}\left( {X, L}\right) \neq 0 \) is contained in the image.\n\nConversely, any divisor \( D = \sum {a}_{i}\left\lbrack {Y}_{i}\right\rbrack \in \operatorname{Div}\left( X\right) \) can be written as \( D = \sum {a}_{i}^{... | Yes |
Lemma 2.3.22 The induced map \( \mathcal{O}\left( {-D}\right) \rightarrow {\mathcal{O}}_{X} \) is injective and the image is the ideal sheaf \( \mathcal{I} \) of \( Y \subset X \) of holomorphic functions vanishing on \( Y \) . | Proof. This is a local statement. Thus, we may assume that \( \mathcal{O}\left( {-Y}\right) \) is trivial and that the map \( \mathcal{O}\left( {-Y}\right) \rightarrow {\mathcal{O}}_{X} \) is given by multiplication with the equation defining \( Y \) . Clearly, this map is injective.\n\nIf \( x \in X \) is either a smo... | No |
Proposition 2.3.26 Let \( L \) be a holomorphic line bundle on a complex manifold \( X \) and suppose that \( {s}_{0},\ldots ,{s}_{N} \in {H}^{0}\left( {X, L}\right) \) is a basis. Then\n\n\[ \n{\varphi }_{L} : X \smallsetminus \operatorname{Bs}\left( L\right) \rightarrow {\mathbb{P}}^{N},\;x \mapsto \left( {{s}_{0}\le... | Proof. Since \( \operatorname{Bs}\left( L\right) = Z\left( {s}_{0}\right) \cap \ldots \cap Z\left( {s}_{N}\right) \subset X \) is a closed subset, the subset \( X \smallsetminus \operatorname{Bs}\left( L\right) \) is an open submanifold of \( X \) . Thus, the notion of a holomorphic map makes sense.\n\nLet us now expla... | Yes |
Proposition 2.3.30 If \( D \) is a principal divisor on a compact curve then \( \deg \left( D\right) = 0 \) . | In order to prove the proposition we will need a lemma. Before stating it let us recall that any meromorphic function \( f \in K\left( X\right) \) defines a holomorphic map \( f : X \smallsetminus P\left( f\right) \rightarrow \mathbb{C} \) . By abuse of notation we denote by \( P\left( f\right) \) not only the pole div... | No |
Lemma 2.3.31 Let \( f \in K\left( X\right) \) be a meromorphic function on a curve \( X \). Then the induced map \( f : X \smallsetminus P\left( f\right) \rightarrow \mathbb{C} \) extends naturally to a holomorphic \( \operatorname{map}X \rightarrow {\mathbb{P}}^{1} \). | Proof. Let \( X = \bigcup {U}_{i} \) be an open covering such that \( {\left. f\right| }_{{U}_{i}} \) is given as \( {g}_{i}/{h}_{i} \) with \( {g}_{i},{h}_{i} \in \mathcal{O}\left( {U}_{i}\right) \). We may assume that \( {g}_{i} \) and \( {h}_{i} \) have no common zero. Indeed, if \( g, h : U \rightarrow \mathbb{C} \... | Yes |
Proposition 2.3.34 For a compact connected curve \( X \) the following conditions are equivalent:\n\n i) The Abel-Jacobi map is not injective.\n\n ii) The curve \( X \) is isomorphic to \( {\mathbb{P}}^{1} \) .\n\n iii) There exist points \( {x}_{1} \neq {x}_{2} \in X \) such that \( \mathcal{O}\left( {x}_{1}\right) \c... | Proof. If \( \mathcal{O}\left( {x}_{1}\right) \cong \mathcal{O}\left( {x}_{2}\right) \), then the line bundle \( L \mathrel{\text{:=}} \mathcal{O}\left( {x}_{i}\right) \) admits two holomorphic sections \( {s}_{1},{s}_{2} \in {H}^{0}\left( {X, L}\right) \) with \( Z\left( {s}_{i}\right) = {x}_{i} \) . Thus, the induced... | Yes |
Proposition 2.4.1 For \( k \geq 0 \) the space \( {H}^{0}\left( {{\mathbb{P}}^{n},\mathcal{O}\left( k\right) }\right) \) is canonically isomorphic to the space \( \mathbb{C}{\left\lbrack {z}_{0},\ldots ,{z}_{n}\right\rbrack }_{k} \) of all homogeneous polynomials of degree \( k \) . | Proof. Note that the section associated to a non-trivial polynomial \( 0 \neq s \in \) \( \mathbb{C}{\left\lbrack {z}_{0},\ldots ,{z}_{n}\right\rbrack }_{k} \) is not trivial. Indeed, the composition \( \mathcal{O}\left( {-1}\right) \subset {\mathbb{P}}^{n} \times {\mathbb{C}}^{n + 1} \rightarrow \) \( {\mathbb{C}}^{n ... | Yes |
Proposition 2.4.3 The canonical bundle \( {K}_{{\mathbb{P}}^{n}} \) is isomorphic to \( \mathcal{O}\left( {-n - 1}\right) \) . | Proof. By definition \( {K}_{{\mathbb{P}}^{n}} \cong \det \left( {\mathcal{T}}_{{\mathbb{P}}^{n}}^{ * }\right) \), where the holomorphic tangent bundle \( {\mathcal{T}}_{{\mathbb{P}}^{n}} \) is uniquely determined by its cocycle \( \left\{ {J\left( {\varphi }_{ij}\right) \circ {\varphi }_{j} : {U}_{i} \cap {U}_{j} \rig... | Yes |
Corollary 2.4.6 One has \( \operatorname{kod}\left( {\mathbb{P}}^{n}\right) = - \infty \) . | Proof. Since the line bundles \( \mathcal{O}\left( d\right) \) have global sections for \( d \geq 0 \), none of the powers \( {K}_{{\mathbb{P}}^{n}}^{\otimes m} = \mathcal{O}\left( {-m\left( {n + 1}\right) }\right) \) for \( m > 0 \) admits non-trivial global sections (Corollary 2.4.2). Thus, the canonical ring \( R\le... | Yes |
Example 2.5.1 Blow-up of a point. Recall that the line bundle \( \mathcal{O}\left( {-1}\right) \) on \( {\mathbb{P}}^{n} \) is given as the incidence variety | Thus, the fibre of the projection \( \pi : \mathcal{O}\left( {-1}\right) \rightarrow {\mathbb{P}}^{n} \) over a line \( \ell \in {\mathbb{P}}^{n} \) is just the line \( \ell \) itself. Let us consider the other projection \( \sigma : \mathcal{O}\left( {-1}\right) \rightarrow {\mathbb{C}}^{n + 1} \) . For \( z \neq 0 \)... | Yes |
Example 2.5.2 Blow-up along a linear subspace. Let \( {\mathbb{C}}^{m} \subset {\mathbb{C}}^{n} \) be the linear subspace satisfying \( {z}_{m + 1} = \ldots = {z}_{n} = 0 \) and denote by \( \left( {{x}_{m + 1} : \ldots : {x}_{n}}\right) \) the homogeneous coordinates of \( {\mathbb{P}}^{n - m - 1} \) . We define \[ {\... | Using the projection \( \pi : {\operatorname{Bl}}_{{\mathbb{C}}^{m}}\left( {\mathbb{C}}^{n}\right) \rightarrow {\mathbb{P}}^{n - m - 1} \) one realizes \( {\operatorname{Bl}}_{{\mathbb{C}}^{m}}\left( {\mathbb{C}}^{n}\right) \) as a \( {\mathbb{C}}^{m + 1} \) -bundle over \( {\mathbb{P}}^{n - m - 1} \) . The fibre over ... | Yes |
Proposition 2.5.5 The canonical bundle \( {K}_{\widehat{X}} \) of the blow-up \( \widehat{X} \) is isomorphic to \( {\sigma }^{ * }{K}_{X} \otimes {\mathcal{O}}_{\widehat{X}}\left( {\left( {n - 1}\right) E}\right) \) . | Proof. The morphism \( \sigma : \widehat{X} \rightarrow X \) induces a sheaf homomorphism \( {\mathcal{T}}_{\widehat{X}} \rightarrow \) \( {\sigma }^{ * }{\mathcal{T}}_{X} \) (either use Exercise 2.2.10 or the discussion in Section 2.6). Taking the determinant and dualizing yields an injection \( {\sigma }^{ * }{K}_{X}... | Yes |
Corollary 2.5.6 For the exceptional divisor \( E = {\mathbb{P}}^{n - 1} \subset \widehat{X} \rightarrow X \) one has \( {\left. \mathcal{O}\left( E\right) \right| }_{E} \cong \mathcal{O}\left( {-1}\right) \) | Proof. Indeed, Propositions 2.4.3 and 2.4.7 yield \( \mathcal{O}\left( {-n}\right) \cong {K}_{{\mathbb{P}}^{n - 1}} \cong \left( {{K}_{\widehat{X}} \otimes }\right. \) \( {\left. \mathcal{O}\left( E\right) )\right| }_{E} \) . Hence, \( {\left. \mathcal{O}\left( nE\right) \right| }_{E} \cong \mathcal{O}\left( {-n}\right... | Yes |
Proposition 2.5.8 Let \( x \in X \) be a point in a complex manifold \( X \) . Then the blow-up \( {\operatorname{Bl}}_{x}\left( X\right) \) is diffeomorphic as an oriented differentiable manifold to \( X \# {\bar{\mathbb{P}}}^{n} \) . | Proof. The assertion is local, so we may assume that \( X \) is the unit disc \( D = \) \( \left\{ {z \in {\mathbb{C}}^{n}\left| \right| z\parallel < 1}\right\} \) and \( x = 0 \) . By definition \( \widehat{D} = {\operatorname{Bl}}_{0}\left( D\right) = \{ \left( {x, z}\right) \in \) \( \left. {D \times {\mathbb{P}}^{n... | Yes |
Proposition 2.6.2 Any complex manifold \( X \) admits a natural almost complex structure. | Proof. Cover \( X \) by holomorphic charts \( \varphi : {U}_{i} \cong {\varphi }_{i}\left( {U}_{i}\right) \subset {\mathbb{C}}^{n} \) and use the almost complex structure defined in Section 1.3. By Proposition 1.3.2 it neither depends on the chart nor on the atlas in the equivalence class specified by the complex struc... | Yes |
Corollary 2.6.8 There exists a natural direct sum decomposition\n\n\[ \mathop{\bigwedge }\limits_{\mathbb{C}}^{k}X = {\bigoplus }_{p + q = k}\mathop{\bigwedge }\limits^{{p, q}}X\text{ and }{\mathcal{A}}_{X,\mathbb{C}}^{k} = {\bigoplus }_{p + q = k}{\mathcal{A}}_{X}^{p, q}. \]\n\nMoreover, \( \overline{\mathop{\bigwedge... | Proof. As Corollary 1.3.4, these assertions are immediate consequences of Proposition 1.2.8. | No |
Proposition 2.6.10 Let \( f : X \rightarrow Y \) be a holomorphic map between complex manifolds. Then the pull-back of differential forms respects the above decompositions, i.e. it induces natural \( \mathbb{C} \) -linear maps \( {f}^{ * } : {\mathcal{A}}^{p, q}\left( Y\right) \rightarrow {\mathcal{A}}^{p, q}\left( X\r... | Proof. As for any differentiable map \( f : X \rightarrow Y \) there exists the natural pull-back map \( {f}^{ * } : {\mathcal{A}}^{k}\left( Y\right) \rightarrow {\mathcal{A}}^{k}\left( X\right) \) which satisfies \( {f}^{ * } \circ {d}_{Y} = {d}_{X} \circ {f}^{ * } \) .\n\nIf \( f \) is holomorphic, then the pull-back... | No |
Proposition 2.6.11 The space \( {H}^{0}\left( {X,{\Omega }_{X}^{p}}\right) \) of holomorphic p-forms on a complex manifold \( X \) is the subspace \( \left\{ {\alpha \in {\mathcal{A}}^{p,0}\left( X\right) \mid \bar{\partial }\alpha = 0}\right\} \) . | Proof. This is a purely local statement. We use that locally the \( p \) -forms \( d{z}_{{i}_{1}} \land \) \( \ldots \land d{z}_{{i}_{p}} \) provide a basis of the complex vector bundle \( \mathop{\bigwedge }\limits^{{p,0}}X \) as well as a holomorphic(!) basis of the holomorphic vector bundle \( {\Omega }_{X}^{p} \) .... | Yes |
Corollary 2.6.12 Let \( f : X \rightarrow Y \) be a holomorphic map between complex manifolds. Then there exist the following natural maps: i) A sheaf homomorphism \( {\mathcal{T}}_{X} \rightarrow {f}^{ * }{\mathcal{T}}_{Y} \), ii) A sheaf homomorphism \( {f}^{ * }{\Omega }_{Y} \rightarrow {\Omega }_{X} \), and iii) A ... | Proof. Assertion ii) implies the other two. Indeed, dualizing ii) yields i). Taking exterior powers of ii), passing to global sections, and composing with \( {H}^{0}\left( {Y,{\Omega }_{Y}^{p}}\right) \rightarrow {H}^{0}\left( {X,{f}^{ * }{\Omega }_{Y}^{p}}\right) \) yields iii). In order to prove ii), we use the exact... | Yes |
Proposition 2.6.15 Let \( X \) be an almost complex manifold. Then the following two conditions are equivalent:\n\ni) \( {d\alpha } = \partial \left( \alpha \right) + \bar{\partial }\left( \alpha \right) \) for all \( \alpha \in {\mathcal{A}}^{ * }\left( X\right) \) .\n\nii) On \( {\mathcal{A}}^{1,0}\left( X\right) \) ... | Proof. The last assertion is easily proved by reducing to the local situation (cf. Section 1.3).\n\nThe implication i) \( \Rightarrow \) ii) is trivial, since \( d = \partial + \bar{\partial } \) clearly implies \( {\Pi }^{0,2} \circ d = 0 \) on \( {\mathcal{A}}^{1,0}\left( X\right) \) .\n\nConversely, \( d = \partial ... | Yes |
Proposition 2.6.17 An almost complex structure \( I \) is integrable if and only if the Lie bracket of vector fields preserves \( {T}_{X}^{0,1} \), i.e. \( \left\lbrack {{T}_{X}^{0,1},{T}_{X}^{0,1}}\right\rbrack \subset {T}_{X}^{0,1} \) . | Proof. Let \( \alpha \) be a \( \left( {1,0}\right) \) -form and let \( v, w \) be sections of \( {T}^{0,1} \) . Then, using the standard formula for the exterior differential (cf. Appendix A) and the fact that \( \alpha \) vanishes on \( {T}^{0,1} \), one finds\n\n\[\n\left( {d\alpha }\right) \left( {v, w}\right) = v\... | Yes |
If \( I \) is an integrable almost complex structure, then \( {\partial }^{2} = \) \( {\bar{\partial }}^{2} = 0 \) and \( \partial \bar{\partial } = - \bar{\partial }\partial \) . Conversely, if \( {\bar{\partial }}^{2} = 0 \), then \( I \) is integrable. | The first assertion follows directly from \( d = \partial + \bar{\partial } \) (Proposition 2.6.15), \( {d}^{2} = 0 \), and the bidegree decomposition.\n\nConversely, if \( {\bar{\partial }}^{2} = 0 \) we show that \( \left\lbrack {{T}_{X}^{0,1},{T}_{X}^{0,1}}\right\rbrack \subset {T}_{X}^{0,1} \) . For \( v, w \) loca... | Yes |
Lemma 2.6.23 If \( E \) is a holomorphic vector bundle then there exists a natural \( \mathbb{C} \) -linear operator \( {\bar{\partial }}_{E} : {\mathcal{A}}^{p, q}\left( E\right) \rightarrow {\mathcal{A}}^{p, q + 1}\left( E\right) \) with \( {\bar{\partial }}_{E}^{2} = 0 \) and which satisfies the Leibniz rule \( {\ba... | Proof. Fix a local holomorphic trivialization \( s = \left( {{s}_{1},\ldots ,{s}_{r}}\right) \) of \( E \) and write a section \( \alpha \in {\mathcal{A}}^{p, q}\left( E\right) \) as \( \alpha = \sum {\alpha }_{i} \otimes {s}_{i} \) with \( \alpha \in {\mathcal{A}}_{X}^{p, q} \) . Then set\n\n\[ \n{\bar{\partial }}_{E}... | Yes |
Corollary 2.6.25 \( {H}^{p, q}\left( {X, E}\right) \cong {H}^{q}\left( {X, E \otimes {\Omega }_{X}^{p}}\right) \) . | Proof. The complex of sheaves \( {\mathcal{A}}^{p,0}\left( E\right) \rightarrow {\mathcal{A}}^{p,1}\left( E\right) \rightarrow {\mathcal{A}}^{p,2}\left( E\right) \rightarrow \ldots \) is a resolution of \( E \otimes {\Omega }_{X}^{p} \) and the sheaves \( {\mathcal{A}}^{p, q}\left( E\right) \) are acyclic. | Yes |
Corollary 3.1.8 The set of all Kähler forms on a compact complex manifold \( X \) is an open convex cone in the linear space \( \left\{ {\omega \in {\mathcal{A}}^{1,1}\left( X\right) \cap {\mathcal{A}}^{2}\left( X\right) \mid {d\omega } = 0}\right\} \) . | Proof. The positivity of an hermitian matrix \( \left( {{h}_{ij}\left( x\right) }\right) \) is an open property and, since \( X \) is compact, the set of forms \( \omega \in {\mathcal{A}}^{1,1}\left( X\right) \cap {\mathcal{A}}^{2}\left( X\right) \) that are locally of the form \( \omega = \frac{i}{2}\sum {h}_{ij}d{z}_... | Yes |
Proposition 3.1.10 Let \( g \) be a Kähler metric on a complex manifold \( X \) . Then the restriction \( {\left. g\right| }_{Y} \) to any complex submanifold \( Y \subset X \) is again Kähler. | Proof. Clearly, \( {\left. g\right| }_{Y} \) is again a Riemannian metric on \( Y \) . Since \( {T}_{x}Y \subset {T}_{x}X \) is invariant under the almost complex structure \( I \) for any \( x \in Y \) and the restriction of it to \( {T}_{x}Y \) is the almost complex structure \( {I}_{Y} \) on \( Y \), the metric \( {... | Yes |
Corollary 3.1.11 Any projective manifold is Kähler. | Proof. By definition a projective manifold can be realized as a submanifold of \( {\mathbb{P}}^{n} \). Restricting the Fubini-Study metric yields a Kähler metric. | Yes |
Proposition 3.2.2 Let \( \left( {X, g}\right) \) be a compact hermitian manifold. Then the following decompositions are orthogonal with respect to \( \left( {\cdot , \cdot }\right) \) :\ni) The degree decomposition \( {\mathcal{A}}_{\mathbb{C}}^{ * }\left( X\right) = {\bigoplus }_{k}{\mathcal{A}}_{\mathbb{C}}^{k}\left(... | Proof. The first assertion follows from the definition of \( {g}_{\mathbb{C}} \) . For ii) let \( \alpha \in \) \( {\mathcal{A}}^{p, q}\left( X\right) \) and \( \beta \in {\mathcal{A}}^{{p}^{\prime },{q}^{\prime }}\left( X\right) \) . Then by Lemma 1.2.24 one has \( {g}_{\mathbb{C}}\left( {\alpha ,\beta }\right) \equiv... | Yes |
Lemma 3.2.3 Let \( X \) be a compact hermitian manifold. Then with respect to the hermitian product \( \left( {\cdot , \cdot }\right) \) the operators \( {\partial }^{ * } \) and \( {\bar{\partial }}^{ * } \) are the formal adjoints of \( \partial \) and \( \bar{\partial } \), respectively. | Proof. The proof is literally the same as for \( d \) and \( {d}^{ * } \) . We recall it for \( {\partial }^{ * } \) . Let \( \alpha \in {\mathcal{A}}^{p - 1, q}\left( X\right) \) and \( \beta \in {\mathcal{A}}^{p, q}\left( X\right) \) . By definition\n\n\[ \left( {\partial \alpha ,\beta }\right) = {\int }_{X}{g}_{\mat... | Yes |
Lemma 3.2.5 Let \( \left( {X, g}\right) \) be a compact hermitian manifold \( \left( {X, g}\right) \) . A form \( \alpha \) is \( \bar{\partial } \) -harmonic (resp. \( \partial \) -harmonic) if and only if \( \bar{\partial }\alpha = {\bar{\partial }}^{ * }\alpha = 0 \) (resp. \( \partial \alpha = {\partial }^{ * }\alp... | Proof. The assertion follows from\n\n\[ \left( {{\Delta }_{\bar{\partial }}\left( \alpha \right) ,\alpha }\right) = {\begin{Vmatrix}{\bar{\partial }}^{ * }\left( \alpha \right) \end{Vmatrix}}^{2} + \parallel \bar{\partial }\left( \alpha \right) {\parallel }^{2}. \]\n\nThus, \( {\Delta }_{\bar{\partial }}\left( \alpha \... | Yes |
Proposition 3.2.6 Let \( \left( {X, g}\right) \) be an hermitian manifold, not necessarily compact. Then\n\ni) \( {\mathcal{H}}_{\partial }^{k}\left( {X, g}\right) = {\bigoplus }_{p + q = k}{\mathcal{H}}_{\bar{\partial }}^{p, q}\left( {X, g}\right) \) and \( {\mathcal{H}}_{\partial }^{k}\left( {X, g}\right) = {\bigoplu... | Proof. Let \( \alpha = \sum {\alpha }^{p, q} \) be the bidegree decomposition of a given form \( \alpha \) . Clearly, if \( {\Delta }_{\bar{\partial }}\left( {\alpha }^{p, q}\right) = 0 \) for all \( \left( {p, q}\right) \) then also \( {\Delta }_{\bar{\partial }}\left( \alpha \right) = 0 \) . On the other hand, \( {\D... | Yes |
Theorem 3.2.8 (Hodge decomposition) Let \( \\left( {X, g}\\right) \) be a compact hermitian manifold. Then there exist two natural orthogonal decompositions\n\n\[ \n{\\mathcal{A}}^{p, q}\\left( X\\right) = \\partial {\\mathcal{A}}^{p - 1, q}\\left( X\\right) \\oplus {\\mathcal{H}}_{\\partial }^{p, q}\\left( {X, g}\\rig... | The orthogonality of the decomposition is easy to verify and the last assertion follows from iii) in Proposition 3.1.12. The crucial fact is the existence of the direct sum decomposition. | No |
Corollary 3.2.9 Let \( \\left( {X, g}\\right) \) be a compact hermitian manifold. Then the canonical projection \( {\\mathcal{H}}_{\\bar{\\partial }}^{p, q}\\left( {X, g}\\right) \\rightarrow {H}^{p, q}\\left( X\\right) \) is an isomorphism. | Proof. Since any \( \\alpha \\in {\\mathcal{H}}_{\\bar{\\partial }}^{p, q}\\left( {X, g}\\right) \) is \( \\bar{\\partial } \) -closed, mapping \( \\alpha \) to its Dolbeault cohomology class \( \\left\\lbrack \\alpha \\right\\rbrack \\in {H}^{p, q}\\left( X\\right) \) defines a map \( {\\mathcal{H}}_{\\partial }^{p, q... | Yes |
Corollary 3.2.10 ( \( \partial \bar{\partial } \) -lemma) Let \( X \) be a compact Kähler manifold. Then for a d-closed form \( \alpha \) of type \( \left( {p, q}\right) \) the following conditions are equivalent: i) The form \( \alpha \) is d-exact, i.e. \( \alpha = {d\beta } \) for some \( \beta \in {\mathcal{A}}_{\m... | Proof. We add another equivalent condition: v) The form \( \alpha \) is orthogonal to \( {\mathcal{H}}^{p, q}\left( {X, g}\right) \) for an arbitrary Kähler metric \( g \) on \( X \) . Since \( X \) is Kähler, we don’t have to specify with respect to which differential operator \( \left( {d,\partial \text{, or}\bar{\pa... | Yes |
Corollary 3.2.12 Let \( \left( {X, g}\right) \) be a compact Kähler manifold. Then there exists a decomposition\n\n\[ \n{H}^{k}\left( {X,\mathbb{C}}\right) = {\bigoplus }_{p + q = k}{H}^{p, q}\left( X\right) \n\]\n\nThis decomposition does not depend on the chosen Kähler structure.\n\nMoreover, with respect to complex ... | Proof. The decomposition is induced by\n\n\[ \n{H}^{k}\left( {X,\mathbb{C}}\right) = {\mathcal{H}}^{k}{\left( X, g\right) }_{\mathbb{C}} = {\bigoplus }_{p + q = k}{\mathcal{H}}^{p, q}\left( {X, g}\right) = {\bigoplus }_{p + q = k}{H}^{p, q}\left( X\right) , \n\]\n\nwhich a priori might depend on the Kähler metric \( g ... | Yes |
Lemma 3.3.1 Let \( X \) be a compact Kähler manifold. The two natural maps \( {H}^{k}\left( {X,\mathbb{C}}\right) \rightarrow {H}^{k}\left( {X,{\mathcal{O}}_{X}}\right) \), induced by \( \mathbb{C} \subset {\mathcal{O}}_{X} \), and \( {H}^{k}\left( {X,\mathbb{C}}\right) \rightarrow {H}^{0, k}\left( X\right) \) , given ... | Proof. We use the following commutative diagram that relates the standard acyclic resolutions of \( \mathbb{C} \) and \( {\mathcal{O}}_{X} \) .\n\n\n\nHere, the first vertical map is the natural inclusion \( \mathbb{... | Yes |
Proposition 3.3.2 (Lefschetz theorem on \( \left( {1,1}\right) \) -classes) Let \( X \) be a compact Kähler manifold. Then \( \operatorname{Pic}\left( X\right) \rightarrow {H}^{1,1}\left( {X,\mathbb{Z}}\right) \) is surjective. | Proof. Let \( \alpha = \rho \left( \widetilde{\alpha }\right) \in \operatorname{Im}\left( {{H}^{2}\left( {X,\mathbb{Z}}\right) \overset{\rho }{ \rightarrow }{H}^{2}\left( {X,\mathbb{C}}\right) }\right) \) . Then, with respect to the bidegree decomposition of \( {H}^{2}\left( {X,\mathbb{C}}\right) \), one can write \( \... | Yes |
Corollary 3.3.6 If \( X \) is a compact Kähler manifold, then \( {\operatorname{Pic}}^{0}\left( X\right) \) is in a natural way a complex torus of dimension \( {b}_{1}\left( X\right) \) . | Proof. We use the bidegree decomposition (Corollary 3.2.12): \( {H}^{1}\left( {X,\mathbb{C}}\right) = \) \( {H}^{1,0}\left( X\right) \oplus {H}^{0,1}\left( X\right) \), the fact that \( \overline{{H}^{1,0}\left( X\right) } = {H}^{0,1}\left( X\right) \), and \( {H}^{1}\left( {X,\mathbb{C}}\right) = \) \( {H}^{1}\left( {... | Yes |
Proposition 3.3.8 The Albanese map alb : \( X \rightarrow \operatorname{Alb}\left( X\right) \) is holomorphic and the pull-back of forms induces a bijection\n\n\[ \mathop{\bigwedge }\limits_{0}\operatorname{Alb}\left( X\right) \cong {H}^{0}\left( {\operatorname{Alb}\left( X\right) ,{\Omega }_{\operatorname{Alb}\left( X... | Proof. Writing the integral \( {\int }_{{x}_{0}}^{x}\alpha \) as \( {\int }_{{x}_{0}}^{{x}_{1}}\alpha + {\int }_{{x}_{1}}^{x}\alpha \), the holomorphicity becomes a local question for \( x \) near a reference point \( {x}_{1} \) . The assertion, therefore, is equivalent to the holomorphicity of \( {\int }_{0}^{x}\alpha... | Yes |
Corollary 3.3.10 Let \( \left( {X, g}\right) \) be a compact Kähler manifold. Then the Lefschetz operator \( L \) and its dual \( \Lambda \) define natural operators on cohomology\n\n\[ L : {H}^{p, q}\left( X\right) \rightarrow {H}^{p + 1, q + 1}\left( X\right) \text{ and }\Lambda : {H}^{p, q}\left( X\right) \rightarro... | Proof. This is a consequence of iii), Remark 3.2.7 and the bidgree decomposition 3.2.12. The last assertion follows from the observation that \( L \) on cohomology is given by the exterior product with the Kähler class \( \left\lbrack \omega \right\rbrack \) . | Yes |
Proposition 3.3.13 (Hard Lefschetz theorem) Let \( \\left( {X, g}\\right) \) be a compact Kähler manifold of dimension \( n \) . Then for \( k \\leq n \n\n\[ \n{L}^{n - k} : {H}^{k}\\left( {X,\\mathbb{R}}\\right) \\cong {H}^{{2n} - k}\\left( {X,\\mathbb{R}}\\right) \n\]\n\n(3.1)\n\nand for any \( k \n\n\[ \n{H}^{k}\\le... | Proof. The first assertion is iii) of Remark 3.2.7. For the second assertion one either uses the fact that \( L \) and \( \\Lambda \) define an \( \\mathfrak{{sl}}\\left( 2\\right) \) -representation on \( {\\mathcal{H}}^{ * }\\left( {X, g}\\right) \) and then concludes as in the proof of Proposition 1.2.30 or one appl... | Yes |
Corollary 3.3.14 The Hodge \( * \) -operator on a compact Kähler manifold \( \left( {X, g}\right) \) acts naturally on cohomology \( {H}^{ * }\left( {X,\mathbb{C}}\right) \) inducing isomorphisms \( * : {H}^{p, q}\left( X\right) \cong \) \( {H}^{n - q, n - p}\left( X\right) \) . The action only depends on the Kähler cl... | Proof. The first assertion follows again from Remark 3.2.7, i). By using Weil's formula 1.2.31 the Hodge \( * \) -operator can be described in terms of the Lefschetz operator \( L \) and the Lefschetz decomposition. In particular, everything depends only on the Kähler class (and the complex structure of \( X \) ). | Yes |
Proposition 3.3.15 (Hodge-Riemann bilinear relation) Let \( \\left( {X, g}\\right) \) be a compact Kähler manifold of dimension \( n \) with Kähler class \( \\left\\lbrack \\omega \\right\\rbrack \) and let \( 0 \\neq \) \( \\alpha \\in {H}^{p, q}{\\left( X\\right) }_{\\mathrm{p}} \) . Then\n\n\[ \n{i}^{p - q}{\\left( ... | Proof. Since any \( \\alpha \\in {H}^{p, q}{\\left( X\\right) }_{\\mathrm{p}} \) can be represented by an harmonic form \( \\alpha \\in {\\mathcal{H}}^{p, q}\\left( {X, g}\\right) \) which is primitive at any point \( x \\in X \) (see Remark 3.3.12), the assertion follows from Corollary 1.2.36. (However, since the prim... | No |
Corollary 3.3.16 (Hodge index theorem) Let \( X \) be a compact Kähler surface, then the intersection pairing\n\n\[ \n{H}^{2}\left( {X,\mathbb{R}}\right) \times {H}^{2}\left( {X,\mathbb{R}}\right) \rightarrow \mathbb{R},\;\left( {\alpha ,\beta }\right) \mapsto {\int }_{X}\alpha \land \beta \n\]\n\nhas index \( \left( {... | Proof. By the bidegree decomposition one has the orthogonal splitting\n\n\[ \n{H}^{2}\left( {X,\mathbb{R}}\right) = \left( {\left( {{H}^{2,0}\left( X\right) \oplus {H}^{0,2}\left( X\right) }\right) \cap {H}^{2}\left( {X,\mathbb{R}}\right) }\right) \oplus {H}^{1,1}\left( {X,\mathbb{R}}\right) .\n\]\n\nFor degree reasons... | Yes |
Let us consider the projective space \( {\mathbb{P}}^{n} \) and the Euler sequence twisted by \( \mathcal{O}\left( {-1}\right) \): \[ 0 \rightarrow \mathcal{O}\left( {-1}\right) \rightarrow {\mathcal{O}}^{\oplus n + 1} \rightarrow {\mathcal{T}}_{{\mathbb{P}}^{n}}\left( {-1}\right) \rightarrow 0. \] | The constant standard hermitian structure on \( {\mathcal{O}}^{\oplus n + 1} \) induces canonical hermitian structures \( {h}_{1} \) on \( \mathcal{O}\left( {-1}\right) \) and \( {h}_{2} \) on \( {\mathcal{T}}_{{\mathbb{P}}^{n}}\left( {-1}\right) \) (see the previous example). The hermitian structure \( {h}_{1} \) on \... | No |
Proposition 4.1.4 Every complex vector bundle admits an hermitian metric. | Proof. Choose an open covering \( X = \bigcup {U}_{i} \) trivializing a given vector bundle \( E \) . Then one might glue the constant hermitian structures on the trivial vector bundles \( {U}_{i} \times {\mathbb{C}}^{r} \) over \( {U}_{i} \) by means of a partition of unity. Here, we use that any positive linear combi... | Yes |
Lemma 4.1.12 Let \( \left( {E, h}\right) \) be an hermitian holomorphic vector bundle on a compact hermitian manifold \( \left( {X, g}\right) \) . Then, with respect to \( \left( {\cdot , \cdot }\right) \), the operator \( {\bar{\partial }}_{E}^{ * } \) on \( {\mathcal{A}}^{p, q}\left( {X, E}\right) \) is adjoint to \(... | Proof. By definition, the second assertion follows from the first one which in turn is proved by the following purely formal calculation:\n\nFor \( \alpha \in {\mathcal{A}}^{p, q}\left( {X, E}\right) \) and \( \beta \in {\mathcal{A}}^{p, q + 1}\left( {X, E}\right) \) one has\n\n\[ \left( {\alpha ,{\bar{\partial }}_{E}^... | Yes |
Corollary 4.1.14 The natural projection \( {\mathcal{H}}^{p, q}\left( {X, E}\right) \rightarrow {H}^{p, q}\left( {X, E}\right) \) is bijective. In particular, \( {H}^{p, q}\left( {X, E}\right) \cong {H}^{q}\left( {X, E \otimes {\Omega }_{X}^{p}}\right) \) is finite-dimensional. | Proof. Indeed, as any harmonic section of \( \mathop{\bigwedge }\limits^{{p, q}}X \otimes E \) is \( {\bar{\partial }}_{E} \) -closed, the projection is well-defined. Moreover, the space of \( {\bar{\partial }}_{E} \) -closed forms in \( {\mathcal{A}}^{p, q}\left( {X, E}\right) \) is \( {\bar{\partial }}_{E}{\mathcal{A... | Yes |
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