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Tuesday, October 29, 2019

Hodge theory: Lecture 13

Dolbeault complex. Let M be a complex manifold of dimension n. Then the canonical almost complex structure on TM gives us (TM)_{\mathbb{C}} = T^{1,0}M \oplus T^{0,1}M, and T^{1,0}M turned out to be a holomorphic vector bundle, the holomorphic tangent bundle of M to be more specific. The decomposition is functorial in the sense that for any holomorphic map \phi : M \rightarrow M' between two complex manifolds, its push forward map (TM)_{\mathbb{C}} \rightarrow \phi^{*}(TM')_{\mathbb{C}} restricts to T^{1,0}M \rightarrow \phi^{*}T^{1,0}M'.

Dually, we have (T^{\vee}M)_{\mathbb{C}} = A^{1,0}_{M} \oplus A^{0,1}_{M}, but we also note that (T^{\vee}M)_{\mathbb{C}} \simeq \mathrm{Hom}_{\mathbb{C}}((TM)_{\mathbb{C}}, \mathbb{C}), where at each fiber, we have (v \otimes 1)^{\vee} \mapsto v^{\vee} \otimes 1. Under this isomorphism we have A^{1,0}_{M} \simeq \mathrm{Hom}_{\mathbb{C}}(T^{1,0}M, \mathbb{C}) and A^{0,1}_{M} \simeq \mathrm{Hom}_{\mathbb{C}}(T^{0,1}M, \mathbb{C}). We also have the complexified k-th exterior power \begin{align*}\left(\bigwedge^{k} T^{\vee}M\right)_{\mathbb{C}} &\simeq \bigwedge^{k} (T^{\vee}M)_{\mathbb{C}} \\ &= \bigwedge^{k} (A^{1,0}_{M} \oplus A^{0,1}_{M}) \\ &= \bigoplus_{p+q = k}\left(\bigwedge^{p} A^{1,0}_{M} \otimes_{\mathbb{R}} \bigwedge^{q} A^{0,1}_{M}\right).\end{align*} We shall write A^{p,q}_{M} := \bigwedge^{p} A^{1,0}_{M} \otimes_{\mathbb{R}} \bigwedge^{q} A^{0,1}_{M}, and call it (p,q)-th exterior power.

Sections of the exterior power \bigwedge^{k} T^{\vee}M, which are called (real) differential k-forms on M, form a sheaf \mathscr{A}_{M}^{k}, and we complexify this sheaf, meaning \mathscr{A}_{M, \mathbb{C}}^{k} := \mathscr{A}_{M}^{k} \otimes_{\underline{\mathbb{R}}} \underline{\mathbb{C}}, and this matches the sheaf of the complexified k-exterior of M, so we have \mathscr{A}_{M, \mathbb{C}}^{k} = \bigoplus_{p+q = k}\left(\bigwedge^{p} \mathscr{A}^{1,0}_{M} \otimes_{\mathscr{C}^{\infty}_{M}} \bigwedge^{q} \mathscr{A}^{0,1}_{M}\right), where \mathscr{A}^{1,0}_{M} is the sheaf of sections of A^{1,0}_{M}, while \mathscr{A}^{0,1}_{M} is the sheaf of sections of A^{0,1}_{M}. We write \mathscr{A}^{p,q}_{M} := \bigwedge^{p} \mathscr{A}^{1,0}_{M} \otimes_{\otimes_{\mathscr{C}^{\infty}_{M}}} \bigwedge^{q} \mathscr{A}^{0,1}_{M} and say the sections of them are (p,q)-sections of M. Note that \overline{\mathscr{A}_{M}^{p,q}} = \mathscr{A}_{M}^{q,p}, where the conjugation happen in the target, namely the (complexified) k-th exterior power. Denote by \Omega^{p}_{M} the sheaf holomorphic sections of the holomorphic vector bundle A^{p,0}. We call the sections of \Omega^{p}_{M} holomorphic p-forms on M Note that we have \Omega^{p}_{M} \hookrightarrow \mathscr{A}^{p,0}_{M}, where the right-hand side is the sheaf of smooth (not necessarily holomorphic) sections of the holomorphic vector bundle A^{p,0}. Note that A^{1,0} is identified to be the (complex) dual of T^{1,0}M, which is a holomorphic vector bundle whose local frame is given by dz_{1}, \dots, dz_{n} for each local chart z_{1}, \dots, z_{n} for M. With such a local chart, say on an open subset U \subset M, we note that \mathscr{A}^{p,q}_{U} is a free \mathscr{C}^{\infty}_{U, \mathbb{C}}-module with basis dz_{I} \wedge d\bar{z_{J}} with I = (i_{1} < \cdots < i_{p}) and J = (j_{1} < \cdots < j_{q}).

Consider the de Rham differential d : \mathscr{A}^{k}_{M} \rightarrow \mathscr{A}^{k+1}_{M}. That is, on each open subset U \subset M, the map d : \mathscr{A}^{k}_{M}(U) \rightarrow \mathscr{A}^{k+1}_{M}(U) is given by the exterior derivative. Considering the complexified version \mathscr{A}^{k}_{M, \mathbb{C}} \overset{d}{\longrightarrow} \mathscr{A}^{k+1}_{M, \mathbb{C}}. For any p, q \geq 0 such that p + q = k, this gives \mathscr{A}^{p,q} \hookrightarrow \mathscr{A}^{k}_{M, \mathbb{C}} \overset{d}{\longrightarrow} \mathscr{A}^{k+1}_{M, \mathbb{C}} \twoheadrightarrow \mathscr{A}^{p+1,q}, which we call \partial^{p,q} and \mathscr{A}^{p,q} \hookrightarrow \mathscr{A}^{k}_{M, \mathbb{C}} \overset{d}{\longrightarrow} \mathscr{A}^{k+1}_{M, \mathbb{C}} \twoheadrightarrow \mathscr{A}^{p,q+1}, which we call \bar{\partial}^{p,q}.

Proposition. When M is complex manifold, for p + q = k, we have d = \partial + \bar{\partial} as a map \mathscr{A}^{p,q}_{M} \overset{d}{\longrightarrow} \mathscr{A}^{p+1,q}_{M} \oplus \mathscr{A}^{p,q+1}_{M}.
Personal remark. We don't really need this because the identity can be checked at the stalks, but one can check that (\mathscr{A}^{k} \otimes_{\underline{\mathbb{R}}} \underline{\mathbb{C}})(U) = \mathscr{A}^{k}(U) \otimes_{\mathbb{R}} \mathbb{C}, even without sheafification after tensoring.

Proof. We suppress M in the notations for sheaves for simplicity. We can check the identity at an open subset U \subset M with the coordinate functions x_{1}, \dots, x_{n}, y_{1}, \dots, y_{n}, which induces the coordinate functions z_{1}, \dots, z_{n}, \bar{z_{1}}, \dots, \bar{z_{n}}. Fix any \omega \in \mathscr{A}^{p,q}(U). Our goal is to show that d\omega = \partial\omega + \bar{\partial}\omega \in \mathscr{A}^{k+1}(U) \otimes_{\mathbb{R}} \mathbb{C}. We may assume that \omega = f dz_{I} \wedge d\bar{z_{J}} with |I| = p and |J| = 1 because \mathscr{A}^{p,q}(U) is \mathbb{C}-linearly generated by such k-forms. We have df = \sum_{j=1}^{n} \frac{\partial f}{\partial z_{j}} dz_{j} + \sum_{j=1}^{n} \frac{\partial f}{\bar{\partial z_{j}}} d\bar{z_{j}} because f is holomorphic. By definition, we have \partial f = \sum_{j=1}^{n} \frac{\partial f}{\partial z_{j}} dz_{j} \in \mathscr{A}^{1,0}(U) and \bar{\partial} f = \sum_{j=1}^{n} \frac{\partial f}{\partial \bar{z_{j}}} d\bar{z_{j}} \in \mathscr{A}^{0,1}(U). Thus, we have \begin{align*}d \omega &= df \wedge dz_{I} \wedge d\bar{z_{J}} \\ &= \partial f \wedge dz_{I} \wedge d\bar{z_{J}} + \bar{\partial} f \wedge dz_{I} \wedge d\bar{z_{J}} \\ &= \partial \omega + \bar{\partial} \omega \end{align*}. This finishes the proof. \Box

As in the above proof, we shall continue drop M in the notations for sheaves, unless it causes too much confusion.

Corollary. Assuming the previous notations, at every (p, q) level, we have

  • \partial^{2} = 0 : \mathscr{A}^{p,q} \rightarrow \mathscr{A}^{p+2,q};
  • \bar{\partial}^{2} = 0 : \mathscr{A}^{p,q} \rightarrow \mathscr{A}^{p,q+2};
  • \partial\bar{\partial} + \bar{\partial}\partial = 0 : \mathscr{A}^{p,q} \rightarrow \mathscr{A}^{p+1,q+1}.

Proof. We have 0 = d^{2} = (\partial + \bar{\partial})^{2} = \partial^{2} + (\partial\bar{\partial} + \bar{\partial}\partial) + \bar{\partial}^{2}. Looking at the different degrees in the grading, the result follows. \Box

Derivation property for \partial and \bar{\partial}. Recall that for \omega \in \mathscr{A}^{k}(U) and \eta \in \mathscr{A}^{l}(U), we have d(\omega \wedge \eta) = d\omega \wedge \eta + (-1)^{k} \omega \wedge d \eta. Thus, the same formula holds for the complexification. This implies that \partial(\omega \wedge \eta) = \partial\omega \wedge \eta + (-1)^{k} \omega \wedge \partial \eta and \bar{\partial}(\omega \wedge \eta) = \bar{\partial}\omega \wedge \eta + (-1)^{k} \omega \wedge \bar{\partial} \eta, for any \omega \in \mathscr{A}^{p+q}(U) and \eta \in \mathscr{A}^{p'+q'}(U) such that p + q = k.

Dolbeault complex. Given a complex manifold M of dimension n, for every p \in \mathbb{Z}_{\geq 0}, we have the following complex of sheaves: 0 \rightarrow \mathscr{A}^{p,0} \overset{\bar{\partial}}{\longrightarrow} \mathscr{A}^{p,1} \overset{\bar{\partial}}{\longrightarrow} \cdots \overset{\bar{\partial}}{\longrightarrow} \mathscr{A}^{p,n-1} \overset{\bar{\partial}}{\longrightarrow} \mathscr{A}^{p,n} \rightarrow 0, taking the global sections, we have the following complex of \mathbb{C}-vector spaces: 0 \rightarrow \Gamma(M, \mathscr{A}^{p,0}) \overset{\bar{\partial}}{\longrightarrow} \Gamma(M, \mathscr{A}^{p,1}) \overset{\bar{\partial}}{\longrightarrow} \cdots \overset{\bar{\partial}}{\longrightarrow} \Gamma(M, \mathscr{A}^{p,n-1}) \overset{\bar{\partial}}{\longrightarrow} \Gamma(M, \mathscr{A}^{p,n}) \rightarrow 0, which is called the p-th Dolbeault complex of M. The q-th cohomology of p-th Dolbeault complex is called the Dolbeault cohomology group of degree (p, q). For notation, we write H^{p,q}(M) := H^{q}(\Gamma(M, \mathscr{A}^{p, \bullet}). We now argue that we can compute the cohomology of the sheaf \Omega^{p}_{M} of holomorphic p-forms on M by the p-th Dolbeault complex.

Lemma 1. We have \ker(\mathscr{A}^{p,0}_{M} \overset{\bar{\partial}}{\longrightarrow} \mathscr{A}^{p,1}_{M}) = \Omega^{p}_{M}, for any p \geq 0.

Proof. We may prove this locally on open subsets U \subset M with chart maps. An element of \mathscr{A}^{p,0}(U) looks like \omega = \sum_{|I| = p} f_{I} dz_{I} where f_{I} \in \mathscr{C}^{\infty}(U). We have \bar{\partial}\omega = \sum_{|I| = p} \sum_{j=1}^{n} \frac{\partial f_{I}}{\partial \bar{z_{j}}} d\bar{z_{j}} \wedge dz_{I}. On U, saying that \omega is holomorphic is equivalent to f_{I} are holomorphic for all I which is precisely when all \partial f_{I} / (\partial \bar{z_{j}}) = 0 for all I and j. This is precisely the same as saying that \bar{\partial} \omega = 0, and this finishes the proof. \Box

Lemma 2 (\bar{\partial}-lemma). Let U \subset X be a nonempty open subset. Fix p \geq 0. Given any q \geq 1 and any smooth (p, q)-form \omega such that \bar{\partial}\omega = 0 on U. Then we may find a local section \beta \in \Gamma(V, \mathscr{A}^{p, q-1}) such that \bar{\partial} \beta = \omega on a nonempty open subset V \subset U.

The point of Lemma 1 and Lemma 2 is that we now have a resolution (i.e., everywhere exact): 0 \rightarrow \Omega^{p}_{M} \rightarrow \mathscr{A}^{p,0}_{M} \rightarrow \cdots \rightarrow \mathscr{A}^{p,n}_{M} \rightarrow 0. What is this good for? We need one more lemma to see this.

Lemma 3. Every \mathscr{A}^{p,q}_{M} is acyclic. Namely, we have H^{i}(X, \mathscr{A}^{p,q}) = 0 for all i \geq 1.

Hence, by a previous posing that says any acyclic resolution of a sheaf computes its cohomology, we have:

Corollary. We have H^{p,q}(M) = H^{q}(M, \Omega^{p}_{M}) \simeq H^{q}(\Gamma(M, \mathscr{A}^{p, \bullet})), where the first equality is just a definition.

We will discuss the proof of Lemma 2 next time, and Lemma 3 later.

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