Grauert's criterion of ampleness

Posted on October 1, 2015 by Dima

tags: complex geometry, projectivity

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

Theorem (Serre vanishing theorem). Let $$X$$ be a compact complex manifold, and $$F$$ be a sheaf. Suppose $$L$$ is a line bundle that admits a hermitian metric with positive curvature (or is ample). Then there exists $$m$$ such that $H^i(X, F \otimes L^n) =0, \textrm{ for all }i > 0, n \geq m$

Theorem. A line bundle $${\cal L}$$ on a complex variety is ample if and only if for any subvariety $$Y$$ of positive dimension there exists a section of $${\cal L}^{\otimes n}|_Y$$ which vanishes at some point but does not vanish identically.

(references: Kleiman. Towards a numerical theory of ampleness or Cartier. Diviseurs amples. I have found them in this Mathoverflow discussion)

Note that contrary to Nakai-Moishezon or Kleiman criteria, Grauert’s criterion does not require that $$X$$ be known to be projective.

Proof. The interesting direction is from right to left.

Proceed by induction on dimension of $$X$$. When $$\dim X=1$$, the statement reduces to the fact that a divisor is ample iff it is positive.

Since by assumption $${\cal L}^{\otimes n}$$ admits a global section, $${\cal L}^{\otimes n} \cong {\cal O}(D)$$ for some effective divisor $$D$$. Wlog, assume $$n=1$$. Then, again by assumption, $${\cal L}|_D$$ has a non-zero section that vanishes at some point, and by induction hypothesis is ample.

Consider the short exact sequence $0 \to {\cal O} \to {\cal O}(D) \to {\cal O}(D)|_D \to 0$ Taking into account that $${\cal L} \cong {\cal O}(D)$$ and tensoring with $${\cal L}$$ several times we find that for all $$m$$ the following sequence is exact $0 \to {\cal L}^{\otimes (m-1)} \to {\cal L}^{\otimes m} \to {\cal L}^{\otimes m}|_D \to 0$

Note that $$H^i(X, i_*i^* {\cal L}^{\otimes m}) = H^i(D, i^*{\cal L}^{\otimes m})$$ where $$i: D \to X$$ is a closed embedding, because closed embeddings are affine (I guess I should say just acyclic’’ in complex analytic setting?). As $${\cal L}^{\otimes m}|_D$$ is ample, the degree $$> 1$$ cohomology of sufficiently big its tensor power vanishes by Serre vanishing and GAGA. Therefore $$\dim H^1(X, {\cal L}^{\otimes m})$$ decreases as $$m$$ increases, and becomes stationary after certain $$m_0$$. Then in the sequence $\ldots H^0(X, {\cal L}^{\otimes m}) \to H^0(X,{\cal L}^{\otimes m}|_D) \to H^1(X, {\cal L}^{\otimes (m-1)} \to H^1(X, {\cal L}^{\otimes m}) \to H^1(X, {\cal L}^{\otimes m}|_D) \to \ldots$ we can replace three last terms by 0. Then the first map is surjective which implies that $${\cal L}^{\otimes m}$$ is generated by global sections.

Here’s why. Suppose $$x \notin D$$. Then the stalk of sections of $$\cal L$$ at $$x$$ is generated by the global section of $$\cal L$$ that defines $$mD$$. If $$x \in D$$ then any generator of the stalk of $${\cal L}^{\otimes m}|_D$$ at $$x$$ is generated by a section of $${\cal L}^{\otimes m}$$ on $$D$$, by assumption. Because the map $$H^0(X, {\cal L}^{\otimes m}) \to H^0(X,{\cal L}^{\otimes}|_D)$$ is surjective this generator can be extended to $$X$$.

Let $$f: X \to \mathbb{P}^k$$ be the map corresponding to $${\cal L}$$. Since $${\cal L}$$ is not trivial, the map is not constant. Let $$Y = f^{-1}(z_0)$$ for some $$z_0 \in Im f$$. Then $$\dim Y < \dim X$$ and by induction, $${\cal L}_Y$$ is ample. On the other hand, $${\cal L}_Y$$ is trivial by construction of $$f$$. Therefore $$f$$ is a finite map, therefore some power of $${\cal L}$$ is very ample.