dd^c lemma

Posted on May 4, 2016 by Dima

In this note I will prove the easy local case of \(dd^c\) lemma.

Let \((M,I)\) be a complex manifold. Extend the action of the complex structure on exterior powers of the complexified tangent bundle forms by \[ \mathbf{I}: \bigwedge{}^* M \to \bigwedge{}^* M, \mathbf{I} := \sum i^{p-q} \cdot \Pi^{p,q} \] where \(\Pi^{p,q}: \bigwedge^{p+q} \to \bigwedge^{p,q}\) is the natural projection. Define a dwisted differential \(d^c = I \circ d \circ I^{-1}\). On the cotangent bundle define it to be \(\mathbf{I}(\alpha)(v_1, \ldots, v_n) := \alpha(\mathbf{I}v_1, \ldots, \mathbf{I}v_n)\).

Proposition. \(\partial = \dfrac{d + i \cdot d^c}{2}, \bar \partial = \dfrac{d - i \cdot d^c}{2}\), where \(\partial\) and \(\bar\partial\) are projections of \(d\) on \((\cdot +1, \cdot)\) and \((\cdot, \cdot+1)\) components.

Proof. Indeed \[ \partial + \bar\partial + i \mathbf{I}^{-1} (\partial + \bar\partial) \mathbf{I} = \partial + \bar\partial + i (i^{q-p-1} \partial + i^{q-p+1}\bar\partial)i^{q-p}=\\ = \partial + \bar\partial + \partial + i^2\bar\partial=2\partial \] Similarly for the second equality. \(\square\)

In particular, \(\partial\bar\partial = -\dfrac{i}{2}dd^c\) on a complex manifold.

Lemma (Poincaré lemma). If \(\alpha\) is a closed form on a polydisc then it is exact.

Lemma (Poincaré-Dolbeault-Grothendieck lemma). If \(\alpha\) is a \(\partial\)-closed, not holomorphic (i.e. \(\alpha \notin A^{n,0} M\)), form on a polydisc then it is \(\bar\partial\)-exact.

(this lemma means in particular that Dolbault resoltions of sheaves of holomorphic forms are acyclic)

Note that since \(\bar(\partial\alpha) = \bar\partial \bar\alpha\), the PDG lemma also holds for \(\partial\).

Lemma (local \(dd^c\) lemma). Let \(\eta\) be a \((1,1)\)-form on a polydisc. The following are equvalent

\(\eta\) is exact, or
\(\eta\) is \(\partial\)-exact, \(\bar\partial\)-closed.
there exists a function \(\chi\) such that \(\eta = dd^c \chi\).

Proof. \(3 \Rightarrow 1\) is immedate.

\(1 \Rightarrow 2\). Recall that \(d^2=\partial^2=\bar\partial^2=0\), where \(d=\partial+\bar\partial\), and therefore \(\bar\partial\partial=-\partial\bar\partial\). If \(\eta = d\alpha\) then there exist forms \(\beta \in A^{1,0}, \gamma\in A^{0,1}\) such that \(\eta = \partial\beta + \bar\partial\gamma\) and \(\bar\partial\beta=\partial\gamma=0\). Then \(\partial(\eta) = \partial^2 \beta + \partial\bar\partial\gamma=-\bar\partial\partial\gamma=0\). Similarly for \(\bar\partial\eta\). Apply PDG and \(\partial\)-PDG to get \(\partial\) and \(\bar\partial\)-exactness of \(\eta\).

\(2 \Rightarrow 3\). By Poincaré-Dolbeault-Grothendieck lemma, there exists a form \(\alpha \in A^{1,0}\) such that \(\bar\partial\alpha=\eta\).

Observe that \(\partial\alpha\) is a closed form: \((\partial + \bar\partial)(\partial\alpha) = \partial^2 \alpha - \partial\bar\partial\alpha=0 -\partial\eta=0\). By Poincar'e lemma, there exists a form \(\alpha'\) such that \(d\alpha' = \partial\alpha' + \bar\partial\alpha' = \partial\alpha\). By grading considerations, \(\bar\partial\alpha' = 0\).

Consider the form \(\beta = \alpha - \alpha' \in A^{1,0}\). Then \(\partial\beta = \partial\alpha - \partial\alpha' = 0\) and \(\bar\partial\beta = \bar\partial\alpha - \bar\partial\alpha' = \eta\). By \(\partial\)-PDG, there exists a function \(\psi\) such that \(\partial\psi = \beta\), so \(\bar\partial\partial\psi=\eta\). Put \(\chi=-i\psi\). \(\square\)

Interestingly, this statement is true globally, i.e. for global \((p,q)\)-forms on a manifold \(M\), if the manifold is Kähler.

Lemma (global \(dd^c\) lemma). Let \(\alpha\) be a form on a Kähler manifold, and assume that \(\eta\) is either \(d\)-, \(\partial\)- or \(\bar\partial\)-exact. Then \(\alpha\) is \(dd^c\)-exact (or, which is the same, \(\partial\bar\partial\)-exact).

This result requires Hodge theory for K"ahler manifolds.

(comments)

The Picard scheme

Posted on April 24, 2016 by Dima

tags: moduli, Abelian varieties

Functors \(Pic\) and \(Div\)

We will denote base change with a subscript: \(X_T = X \times T\).

If \(X\) is a scheme, the Picard group of \(X\) is defined to be the group of isomorphism classes of invertible sheaves on \(X\). The relative Picard functor of an \(S\)-scheme \(X\) is defined as \[ \operatorname{Pic}_{X/S}(T) := \operatorname{Pic}(X_T) / \operatorname{Pic}(T) \] where the embedding \(\operatorname{Pic}(T) \hookrightarrow \operatorname{Pic}(X_T)\) is given by the pullback along the structure maps \(X_T \to T\).

Caveat: all representability results work with a sheafification of this functor in some topology, Zariski, étale, or fppf. Unless \(X \to S\) is proper and has a section, they need not be isomorphic.

An effective divisor is a closed subscheme such that its ideal is invertible. If \(f: X \to S\) is a morphism of schemes then a relative effective divisor on \(X\) is an effective divisor \(D\) such that \(D\) is flat over \(S\). For a morphism \(X \to S\) define the functor of relative divisors \[ \operatorname{Div}_{X/S}(T) := \{\ \textrm{ relative effective divisors on } X_T/T \ \} \]

We are interested in representability of this functor, so to this end we prove a little lemma.

Lemma. Let \(X \to S\) be a flat morphism. Let \(D\) be a closed subscheme of \(X\) flat over \(S\). Then \(D\) is relative effective divisor in a neighbourhood of \(x \in X\) if and only if \(D_s\) is cut out in a neighbourhood of \(x\) in \(X_s\), where \(s\) is the image of \(x\), by a single non-zero element of \(\operatorname{\mathcal{O}}_{X_s,x}\) (which amounts to being an effective divisor in \(X_s\) but we won’t prove it).

Construction of Hilb and Quot

Posted on February 16, 2016 by Dima

tags: moduli

Hilbert polynomial

If \(X\) is a projective variety and \(\mathcal{F}\) is a sheaf on \(X\) then the function \[ \chi(\mathcal F, n) = \sum (-1)^i H^i(X, \mathcal F(n)) \] is called the Hilbert function of \(\mathcal F\).

Lemma (Snapper lemma). It is a polynomial.

Proof. Induction on the dimension of support of \(\mathcal{F}\). Notice that \(\chi\) is additive in extensions, i.e. \(\chi(\mathcal{F}/\mathcal{G}) = \chi(\mathcal{F}) - \chi(\mathcal{G})\), as follows from long exact sequence of cohomology.

Recall that for a divisor \(D\) the sheaf \(\mathcal{O}(D)\) is defined as \[ \mathcal{O}(D)(U) := \{\ f \in \mathcal{K}(U) \ \mid\ (f) + D \geq 0 \ \} \] where \(\mathcal{K}\) is the constant sheaf of rational functions on \(X\). Let \(H\) be a hyperplane in \(\mathbb{P}^n\), one has a natural short exact sequence \[ 0 \to \mathcal{O}_{\mathbb{P}^n}(-H) \to \mathcal{O}_{\mathbb{P}^n} \to \mathcal{O}_H \to 0 \] Assume that \(H\) is chosen so that \(\dim \operatorname{supp}\mathcal{F}\cap H < \dim \operatorname{supp}\mathcal{F}\). Tensoring with \(\mathcal{F}\), taking into account that \(\mathcal{O}_{\mathbb{P}^n}(-H) \cong \mathcal{O}_{\mathbb{P}^n}(-1)\), and passing to the long exact sequence we get \[ 0 \to Tor_1(\mathcal{F}, \mathcal{O}_H) \to \mathcal{F}(-1) \to \mathcal{F}\to \mathcal{F}\otimes \mathcal{O}_H \to 0 \] Note that by looking at long exact cohomolgy sequences one establishes that Hilbert polynomial is an additive invariant. Read more (comments)

Grauert's criterion of ampleness

Posted on October 1, 2015 by Dima

tags: line bundles, positivity

Here’s a writeup of a proof of Grauert’s criterion for ampleness (here is the link to the original paper). Since often proving that a variety is algebraic is not far from proving that it is projective, this criterion can be useful in judging algebraicity of a variety.

We start with some observations on cohomology of invertible sheaves which are powers of the invertible sheaf associated to hyperplane section.

Recall that if \(D\) is a (Cartier) divisor on a variety \(X\) then it gives rise to a sheaf \[ {\cal O}(D) := \{ f \in k(X) \mid (f) + D \geq 0 \} \] and a section \(s \in H^0(X, {\cal O}(D))\) such that \((s) = D\). Let \(V = H^0(X, L)\). Then a (generally speaking, partial) map \(\iota: X \dashrightarrow \mathbb{P}(V^\vee)\) is defined: \(x \mapsto (f \mapsto f(x))\). The value on the right, before projectivization, depends on trivialization, hence is only well-defined up to a constant, but since we projectivize, the map is well-defined. A line bundle (or a divisor \(D\)) is called very ample if \(\iota\) is a closed embedding, ample if some multiple of it is ample.

If \(D\) is ample, then \(H:=mD\) is a hyperplane section for some \(m > 0\). Then \(D_H^r\) is the degree of \(X\), hence positive.

Theorem. Let \(X=\mathbb{P}^d\)

\(H^i(X, {\cal O}(n)) = 0\), if \(0 < i < d\) or \(i < 0\)
\(H^i(X, {\cal O}(n)) \cong H^{n-i}(X, {\cal O}(-d-n-1))^\vee\)

A paranthesis on ampleness.

Lemma. Let \(X\) be a scheme covered by finitely many affine schemes \(X_i\) such that \(X_i\) is the locus of points \(x\) such that \(s_i\) generates \(\operatorname{\mathcal{O}}_{X,x}\) for an \(s_i \in H^0(X, \operatorname{\mathcal{L}})\). Let \(s_{ij} \in H^0(X_i, \operatorname{\mathcal{L}})\) be sections such that \(s_{ij}/s_i\) is a base in \(H^0(X_i, \operatorname{\mathcal{O}}_X)\) for each \(i\). (Note that \(s_i\) generate \(\operatorname{\mathcal{L}}\). ) Then the morphism \(X \to \operatorname{Proj}[s_i, s_{ij}]\) is an embedding.

Theorem. Let \(\operatorname{\mathcal{L}}\) be a sheaf such that for any finitely generated quasi-coherent \(\operatorname{\mathbb{F}}\) there exists an \(n_0\) such that \(\operatorname{\mathbb{F}}\otimes \operatorname{\mathcal{L}}^n\) is generated by global sections for \(n \geq n_0\). Then there exists an \(m\) such that \(\operatorname{\mathcal{L}}^m\) is very ample (defines an embedding into \(P^n\)).

Proof ([Liu, 5.1.34, p.169]). Let \(U\) be an affine neighbourhood of \(x\) such that \(\operatorname{\mathcal{L}}|_U\) is free, and let \(\operatorname{\mathcal{I}}\) be the sheaf of ideals that cuts out the complement of \(U\). There exists an \(n_0\) such that \(\operatorname{\mathcal{I}}\operatorname{\mathcal{L}}^n\) is globally generated, so there is a section \(s \in H^0(X, \operatorname{\mathcal{I}}\operatorname{\mathcal{L}}^n) \subset H^0(X, \operatorname{\mathcal{L}}^n)\) that doesn’t vanish at \(x\). Since sectios of \(\operatorname{\mathcal{I}}\operatorname{\mathcal{L}}^n\) that vanish on the complement to \(U\), \(X_s\) is contained in \(U\).

Now by compactness \(X\) is covered by finitely many affines of the form \(X_{s_i}\), \(s_i \in H^0(X, \operatorname{\mathcal{L}}^n)\) with \(H^0(X_{s_i}, \operatorname{\mathcal{O}})\) finitely generated. We can apply the previous lemma to conclude.