Kodaira Vanishing Theorem

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

§Cohomology of Projective Space, ⁋Proposition 7 (Serre Vanishing)’s Serre vanishing theorem guarantees that for an ample line bundle \(\mathcal{L}\) and a coherent sheaf \(\mathcal{F}\) on a projective variety, \(H^i(X, \mathcal{F} \otimes \mathcal{L}^{\otimes m}) = 0\) for \(i > 0\) when \(m\) is sufficiently large. However, this result is merely an asymptotic property and gives no information about the specific \(m\) at which vanishing begins.

The Kodaira vanishing theorem is a far more refined result: it guarantees that the higher cohomology of the tensor product \(\omega_X \otimes \mathcal{L}\) of the canonical bundle \(\omega_X\) and an ample line bundle \(\mathcal{L}\) always vanishes. In this post we examine the Kodaira vanishing theorem, its applications, and how it is used in algebraic geometry.

Kodaira Vanishing Theorem

The basic setup is as follows. \(X\) is an \(n\)-dimensional smooth projective variety, \(\mathcal{L}\) is an ample line bundle on \(X\), and \(\omega_X = \det \Omega_X^1 = \Omega_X^n\) is the canonical line bundle. (§Canonical Line Bundle, ⁋Definition 5) The Kodaira vanishing theorem can then be stated as follows.

Proposition 1 (Kodaira vanishing) Suppose \(\operatorname{char}\mathbb{K} = 0\), and let \(X\) be an \(n\)-dimensional smooth projective variety with an ample line bundle \(\mathcal{L}\). Then for all \(p > 0\),

\[H^p(X, \omega_X \otimes \mathcal{L}) = 0\]

holds. More generally, for \(p, q\) satisfying \(p + q > n\),

\[H^p(X, \Omega^q \otimes \mathcal{L}) = 0\]

holds.

The first statement is obtained from the second by setting \(q = n\), and we have already seen it in §The Riemann–Roch Theorem for Surfaces, ⁋Proposition 7 (Kodaira Vanishing Theorem). The second statement extends this to arbitrary form degree \(q\) and is called the Akizuki–Nakano vanishing. The proof of this proposition is quite technical, so in this post we focus on how it is used in algebraic geometry rather than giving a rigorous proof.

As the statement shows, Kodaira vanishing kills higher cohomology after twisting by the canonical bundle. Using Serre duality, this can be rewritten as the following equivalent statement.

Proposition 2 Under the hypotheses of Proposition 1 (Kodaira vanishing), for all \(p < n\),

\[H^p(X, \mathcal{L}^{-1}) = 0\]

holds.

Proof

By Serre duality from §Serre Duality,

\[H^p(X, \mathcal{L}^{-1}) \cong H^{n-p}(X, \omega_X \otimes \mathcal{L})^\vee\]

holds. If \(p < n\), then \(n - p > 0\), so the right-hand side is \(0\) by Proposition 1 (Kodaira vanishing).

Since these two formulations are completely equivalent via Serre duality, as seen in the proof above, we may use whichever is more convenient in a given situation.

The simplest nontrivial example to which Kodaira vanishing applies is projective space \(X = \mathbb{P}^n\).

Example 3 From the Euler exact sequence in §Canonical Line Bundle, ⁋Proposition 7 (Euler Exact Sequence), we verified that

\[\omega_{\mathbb{P}^n} \cong \mathcal{O}(-n-1)\]

and in §Line Bundles and Vector Bundles, ⁋Example 12 we checked that any line bundle on \(\mathbb{P}^n\) is of the form \(\mathcal{O}(d)\). Among these, those with \(d > 0\) are ample. Therefore, Kodaira vanishing asserts that

\[H^p(\mathbb{P}^n, \mathcal{O}(d - n - 1)) = 0\]

for all \(d > 0\) and all \(p > 0\).

Since we already know the cohomology of every line bundle from §Cohomology of Projective Space, ⁋Proposition 1 (Bott), we can verify this directly. It states that

\[H^q(\mathbb{P}^n, \mathcal{O}(k)) = \begin{cases} \mathbb{K}[\x_0, \ldots, \x_n]_k & q = 0, k \geq 0 \\ \mathbb{K}[\x_0^{-1}, \ldots, \x_n^{-1}]_{-k-n-1} & q = n, k \leq -n-1 \\ 0 & \text{otherwise} \end{cases}\]

From this, cohomology automatically vanishes for \(q \neq 0, n\), so we need only consider \(q = n\). According to the formula, for this to be nonzero we must have \(k \leq -n-1\). But in our situation \(k = d - n - 1\) with \(d > 0\), so this is impossible, and we recover the Kodaira vanishing theorem.

Applications of the Kodaira Vanishing Theorem

Now, as previewed earlier, we examine applications of the Kodaira vanishing theorem. First, by the Riemann–Roch theorem from the previous post, for a divisor \(D\) on a surface \(S\),

\[\rchi(\mathcal{O}_S(D)) = \frac{1}{2} D \cdot (D - K_S) + \rchi(\mathcal{O}_S)\]

holds. (§The Riemann–Roch Theorem for Surfaces, ⁋Proposition 4 (Riemann–Roch for surfaces)) The power of this formula lies in the fact that \(\rchi\) can be computed purely from algebraic and topological data, but the problem is that \(\rchi\) is the alternating sum of \(h^0, h^1, h^2\). Thus, when we simply want to know \(h^0(S, \mathcal{O}_S(D))\), we must determine the higher cohomology groups separately, so the Riemann–Roch formula alone does not give a direct answer.

To apply the Kodaira vanishing theorem in this situation, suppose \(\mathcal{L} \cong \mathcal{O}_S(L)\) is an ample line bundle. Then we know that

\[\omega_S \otimes \mathcal{L} \cong \mathcal{O}_S(K_S + L)\]

and substituting this above and using \(h^1(S, \omega_S \otimes \mathcal{L}) = h^2(S, \omega_S \otimes \mathcal{L}) = 0\) from Proposition 1 (Kodaira vanishing), we obtain

\[\rchi(S, \omega_S \otimes \mathcal{L}) = h^0(S, \omega_S \otimes \mathcal{L})\]

Hence we can read off \(h^0(S, \omega_S \otimes \mathcal{L})\) immediately from the right-hand side of the Riemann–Roch formula.

Another application is the computation of plurigenera. The plurigenus \(P_m(X)\) of a smooth projective variety \(X\) is a generalization of the geometric genus \(p_g(X)\) and is a birational invariant of surfaces. (§The Riemann–Roch Theorem for Surfaces, ⁋Definition 12) Kodaira vanishing can be used directly to compute these invariants.

For example, for a curve \(C\) we know that its plurigenera are determined by the genus; indeed \(P_m(g)\) is given as a function of \(g\) (and \(m\)). Thus for curves, plurigenera are not particularly interesting invariants. The interesting case arises in higher dimensions such as surfaces, where a birational invariant is no longer determined by a single number and all plurigenera become genuinely necessary.

As seen in §The Riemann–Roch Theorem for Surfaces, for a divisor \(D\) on a surface \(S\) the Riemann–Roch formula is

\[\rchi(\mathcal{O}_S(D)) = \frac{1}{2} D \cdot (D - K_S) + \rchi(\mathcal{O}_S)\]

and to compute plurigenera, using \(\omega_S^{\otimes m} \cong \mathcal{O}_S(mK_S)\) and substituting \(D = mK_S\), we get

\[\rchi(\mathcal{O}_S(mK_S)) = \frac{m(m-1)}{2} K_S^2 + \rchi(\mathcal{O}_S)\]

Now, if \(m \geq 2\) and \(K_S\) is ample, then \((m-1)K_S\) is also ample, so applying Proposition 1 (Kodaira vanishing) to \(mK_S = K_S + (m-1)K_S\) yields \(h^1 = h^2 = 0\). Therefore, from this formula we can directly compute \(P_m(S) = h^0(S, \mathcal{O}_S(mK_S))\).

In this case, moreover, the expression for plurigenera is asymptotically quadratic. This leads to the following definition.

Definition 4 The Kodaira dimension \(\kappa(X)\) of a smooth projective variety \(X\) is defined as follows. If \(P_m(X) = 0\) for all \(m \geq 1\), then \(\kappa(X) = -\infty\). Otherwise, \(\kappa(X)\) is defined as the smallest integer \(\kappa \geq 0\) satisfying \(P_m(X) = O(m^\kappa)\). That is,

\[\kappa(X) = \min\{k \in \mathbb{Z}_{\geq 0} \mid P_m(X) = O(m^k)\}\]

Equivalently, it can also be written as

\[\kappa(X) = \limsup_{m \rightarrow \infty} \frac{\log P_m(X)}{\log m}\]

That the set over which the minimum is taken is nonempty follows from the fact that \(P_m(X) = O(m^{\dim X})\) holds for any smooth projective variety; from this we know that \(\kappa(X)\) is well-defined and that \(\kappa(X) \leq \dim X\) always holds. Hence for surfaces, \(\kappa \in \{-\infty, 0, 1, 2\}\). The Enriques–Kodaira classification classifies surfaces largely by Kodaira dimension, and for the cases \(\kappa = 0\) and \(\kappa = -\infty\) it provides additional detailed classification using the geometric genus \(p_g\) and the irregularity \(q\).

In §Linear Systems, ⁋Definition 9, we defined a line bundle \(\mathcal{L}\) being very ample by the condition that the morphism \(\varphi_{\mathcal{L}}: X \rightarrow \mathbb{P}(\Gamma(X, \mathcal{L}))\) defined by the complete linear system \(\lvert \mathcal{L} \rvert\) is a closed embedding. At that time we did not yet have the language of sheaf cohomology, but now that we have introduced it, we can put it to better use.

First, suppose a very ample line bundle \(\mathcal{L}\) is given, and consider the closed embedding \(\varphi_\mathcal{L}: X \rightarrow \mathbb{P}^N\) it defines. Since \(\varphi\) is an embedding, we know that \(\varphi_\mathcal{L}(p) \neq \varphi_\mathcal{L}(q)\); moreover, since \(\varphi_\mathcal{L}\) is a closed embedding, \(\dd{\varphi_\mathcal{L}}\) is injective, and hence the dual map on cotangent spaces \(\mathfrak{m}_{\varphi_{\mathcal{L}}(p)}/\mathfrak{m}_{\varphi_{\mathcal{L}}(p)}^2 \longrightarrow \mathfrak{m}_p/\mathfrak{m}_p^2\) is surjective. From this we know that the following two conditions hold.

1. \(\varphi_\mathcal{L}\) separates points. That is, for any two distinct closed points \(p, q \in X\), there exists a global section \(s \in H^0(X, \mathcal{L})\) such that \(s(p) = 0\) and \(s(q) \neq 0\).
2. \(\varphi_\mathcal{L}\) separates tangent vectors. That is, for any closed point \(p \in X\), the collection of sections vanishing at \(p\), \(\{ s \in H^0(X, \mathcal{L}) \mid s(p) = 0 \}\), spans the vector space \(\mathfrak{m}_p\mathcal{L}_p / \mathfrak{m}_p^2\mathcal{L}_p\) corresponding to the cotangent space.

The first condition means that the evaluation map

\[H^0(X, \mathcal{L}) \longrightarrow \mathcal{L}_p \oplus \mathcal{L}_q\]

is surjective, and the second condition means that the image of the restriction map by sections vanishing at \(p\),

\[\{s \in H^0(X, \mathcal{L}) \mid s(p) = 0\} \longrightarrow \mathfrak{m}_p\mathcal{L}_p / \mathfrak{m}_p^2\mathcal{L}_p\]

spans all of \(\mathfrak{m}_p\mathcal{L}_p / \mathfrak{m}_p^2\mathcal{L}_p\). It is not difficult to check that the converses also hold. That is, the following holds.

Proposition 5 For a projective variety \(X\) over an algebraically closed field and a line bundle \(\mathcal{L}\) on it, \(\mathcal{L}\) being very ample is equivalent to simultaneously satisfying the two separation conditions above.

Now suppose \(\mathcal{L}\) is an ample line bundle, and let us examine how these separation conditions are verified for \(\mathcal{L}^{\otimes m}\) via cohomology. First, for (1), consider the closed subset \(Z = \{p\} \cup \{q\}\) containing two points \(p \neq q\); for the ideal sheaf \(\mathcal{I}_Z\) defining \(Z\) we obtain the short exact sequence

\[0 \longrightarrow \mathcal{I}_Z \otimes \mathcal{L}^{\otimes m} \longrightarrow \mathcal{L}^{\otimes m} \longrightarrow \mathcal{L}^{\otimes m} \otimes \mathcal{O}_Z \longrightarrow 0\]

Here \(\mathcal{L}^{\otimes m} \otimes \mathcal{O}_Z\) is a line bundle on \(Z\), and

\[H^0(Z, \mathcal{L}^{\otimes m}\rvert_Z) \cong \mathcal{L}^{\otimes m}_p \oplus \mathcal{L}^{\otimes m}_q\]

holds. Considering the induced long exact sequence

\[H^0(X, \mathcal{L}^{\otimes m}) \longrightarrow H^0(Z, \mathcal{L}^{\otimes m}\rvert_Z) \longrightarrow H^1(X, \mathcal{I}_Z \otimes \mathcal{L}^{\otimes m})\]

if \(H^1(X, \mathcal{I}_Z \otimes \mathcal{L}^{\otimes m}) = 0\), then the evaluation map is surjective and separation of points holds.

Similarly, for (2), consider the first infinitesimal neighborhood of the point \(p\), \(\Spec(\mathcal{O}_{X,p}/\mathfrak{m}_p^2)\), and let \(\mathcal{I}_p\) be the ideal sheaf of \(p\); from the short exact sequence

\[0 \longrightarrow \mathcal{I}_p^2 \otimes \mathcal{L}^{\otimes m} \longrightarrow \mathcal{L}^{\otimes m} \longrightarrow \mathcal{L}^{\otimes m} \otimes (\mathcal{O}_X / \mathcal{I}_p^2) \longrightarrow 0\]

the induced long exact sequence

\[H^0(X, \mathcal{L}^{\otimes m}) \longrightarrow H^0(X, \mathcal{L}^{\otimes m} \otimes (\mathcal{O}_X / \mathcal{I}_p^2)) \longrightarrow H^1(X, \mathcal{I}_p^2 \otimes \mathcal{L}^{\otimes m})\]

shows that if \(H^1(X, \mathcal{I}_p^2 \otimes \mathcal{L}^{\otimes m}) = 0\), then separation of tangent vectors holds.

Since \(\mathcal{I}_Z\) and \(\mathcal{I}_p^2\) are coherent sheaves, applying §Cohomology of Projective Space, ⁋Proposition 7 (Serre Vanishing) to \(\mathcal{F} = \mathcal{I}_Z\) and \(\mathcal{F} = \mathcal{I}_p^2\), the two \(H^1\)’s above both vanish for sufficiently large \(m\). Therefore, the sections of \(\mathcal{L}^{\otimes m}\) satisfy both separation conditions, and by Proposition 5, \(\mathcal{L}^{\otimes m}\) is very ample. That is, for an ample line bundle, \(\mathcal{L}^{\otimes m}\) is very ample for all sufficiently large \(m\).

Meanwhile, Kodaira vanishing enters the classical proof of Proposition 6 (Kodaira embedding) in a different way. In that proof, one applies vanishing to a line bundle on the blow-up \(\pi: \widetilde{X} \rightarrow X\) of \(p\) and \(q\) with the twist lowered by the exceptional divisor, so that the object of vanishing becomes a line bundle again, reducing to the form of Proposition 1 (Kodaira vanishing). Furthermore, the condition that \(\mathcal{L}^{\otimes m}\) be not only very ample but also that the embedding it defines be projectively normal can be obtained by verifying the surjectivity of the related multiplication map

\[S^\mu H^0(X, \mathcal{L}^{\otimes m}) \longrightarrow H^0(X, \mathcal{L}^{\otimes \mu m})\]

and what supplies the necessary vanishing together with a concrete range of \(m\) is the Castelnuovo–Mumford regularity from §Cohomology of Projective Space, §§Regularity. Such vanishing guarantees that higher cohomology does not obstruct the generation of sections, allowing one to handle the abundance of linear systems quantitatively.

Kodaira Embedding Theorem

The most famous application of Kodaira vanishing is the Kodaira embedding theorem. However, this ventures into the realm of complex manifolds, so we only briefly introduce it here. First, a compact complex manifold \(X\) is a Kähler manifold when a compatible Riemannian metric, symplectic form, and complex structure are defined on \(X\). In this case, if a Hermitian metric \(h\) is given on a line bundle \(\mathcal{L}\), its curvature form \(\Theta_h\) is defined, and \(\mathcal{L}\) being positive means that \(\frac{i}{2\pi}\Theta_h\) is a positive definite \((1,1)\)-form. Then the following holds.

Proposition 6 (Kodaira embedding) Let \(X\) be a compact Kähler manifold and \(\mathcal{L}\) a positive line bundle. Then for sufficiently large \(k\), \(\mathcal{L}^{\otimes k}\) is very ample, and in particular \(\mathcal{L}\) is an ample line bundle. Hence \(X\) is a projective variety.

That is, using this proposition one can show that a Kähler manifold is a projective variety.

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References

[Hart] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Springer, 1977.
[Laz] R. Lazarsfeld, Positivity in Algebraic Geometry I & II, Ergebnisse der Mathematik, Springer, 2004.
[Kod] K. Kodaira, On a differential-geometric method in the theory of analytic stacks, Proceedings of the National Academy of Sciences, 1953.