*dd*^{c} lemma

^{c}

*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*

## 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*

## 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*

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 \(m\).