# ddc lemma

Posted on May 4, 2016 by Dima

tags: Kahler manifolds

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.

# 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.

Theorem. Let $${\operatorname{\mathcal{L}}}$$ be a line bundle such that for any sheaf of ideals $${\operatorname{\mathcal{I}}}$$ there exists a number $$n$$ such that $$H^1(X, {\operatorname{\mathcal{I}}}\otimes {\operatorname{\mathcal{L}}}^n)$$ vanishes. Then $${\operatorname{\mathcal{L}}}$$ is ample, i.e. $${\operatorname{\mathbb{F}}}\otimes {\operatorname{\mathcal{L}}}^m$$ is globally generated for big enough $$m$$.