Kodaira Vanishing Theorem

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§Cohomology of Projective Spaces, ⁋Proposition 4, the Serre vanishing theorem guarantees that for an ample line bundle \(\mathcal{L}\) and a coherent sheaf \(\mathcal{F}\) on a projective variety, \(H^i(X, \mathcal{F} \otimes \mathcal{L}^{\otimes m}) = 0\) (\(i > 0\)) holds for sufficiently large \(m\). However, this result is merely an asymptotic property and gives no information about exactly which \(m\) the vanishing begins at.

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

Kodaira Vanishing Theorem

The basic setup we consider 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 Bundle, ⁋Definition 1) Then the Kodaira vanishing theorem can be stated as follows.

Proposition 1 (Kodaira vanishing) Let \(X\) be an \(n\)-dimensional smooth projective variety and \(\mathcal{L}\) an ample line bundle. 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 claim is obtained from the second by setting \(q=n\). The proof of this proposition is rather technical, so in this post we focus on how it is used in algebraic geometry rather than giving a rigorous proof.

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

Proposition 2 Under the assumptions of Proposition 1, for all \(p < n\),

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

holds.

Proof

By Serre duality from §Serre Duality, ⁋Proposition 2,

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

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

These two formulations are completely equivalent via Serre duality, as seen in the proof above, so we may use whichever is more convenient depending on the situation.

The simplest nontrivial example of Kodaira vanishing is provided by projective space \(X = \mathbb{P}^n\).

Example 3 In §Canonical Bundle, ⁋Proposition 7 we verified that

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

and in §Line Bundles and Vector Bundles, ⁋Example 16 we verified that any line bundle on \(\mathbb{P}^n\) is of the form \(\mathcal{O}(d)\). Among these, the \(\mathcal{O}(d)\) with \(d>0\) are ample line bundles. Therefore, Kodaira vanishing asserts that the following vanishing

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

holds for all \(d>0\) and all \(i>0\).

Since we know the cohomology of every line bundle from §Cohomology of Projective Spaces, ⁋Proposition 1, we can verify this directly. According to this,

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

and from this all cohomology automatically vanishes for \(q\neq 0\), so our only concern is when \(q=n\). Now, according to the formula above, for this to be nonzero we must have \(k\leq -n-1\). But in our situation \(k=d-n-1\) and \(d>0\), so this is impossible, and therefore we can verify the Kodaira vanishing theorem again.

Applications of the Kodaira Vanishing Theorem

Now, as previewed earlier, we examine applications of the Kodaira vanishing theorem. First, according to the Riemann-Roch theorem in the previous post, for a divisor \(D\) on a surface \(S\) we have

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

(§Riemann-Roch for Surfaces, ⁋Proposition 4). 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 values of the higher cohomology groups separately, so the Riemann-Roch formula alone does not yield a direct answer.

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

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

and substituting this above gives

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

and therefore computing only the right-hand side of the Riemann-Roch formula immediately yields \(h^0(S, \omega_S \otimes \mathcal{L})\).

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. (§Riemann-Roch for Surfaces, ⁋Definition 12) Kodaira vanishing can be used directly in computing these invariants.

For example, in the case of a curve \(C\), we know that its birational class is determined by the genus, and the plurigenus \(P_m(g)\) is given as a function of \(g\) (and \(m\)). That is, for a curve \(C\), the plurigenus is not essentially an interesting invariant. This becomes interesting in higher dimensions such as surfaces, where birational invariants are not determined by a single number and all plurigenera become genuinely necessary.

As seen in §Riemann-Roch for Surfaces, for a divisor \(D\) on a surface \(S\) the Riemann-Roch formula is given by

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

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

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

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

On the other hand, in this case the plurigenera can be thought of as asymptotically quadratic. This leads to

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,

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

Equivalently, it can also be written as

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

From the above computation, we know that for surfaces \(\kappa \in \{-\infty, 0, 1, 2\}\). The Enriques–Kodaira classification classifies surfaces broadly by Kodaira dimension, and for the cases \(\kappa=0\) and \(\kappa=-\infty\) provides additional detailed classification using the geometric genus \(p_g\) and irregularity \(q\).

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

First, suppose a very ample line bundle \(\mathcal{L}\) is given, and consider the closed embedding \(\varphi_\mathcal{L}: X\rightarrow \mathbb{P}^N\) defined by it. Then, since \(\varphi\) is an embedding, we know that \(\varphi_\mathcal{L}(p)\neq \varphi_\mathcal{L}(q)\), and moreover, since \(\varphi_\mathcal{L}\) is a closed embedding, \(d\varphi_\mathcal{L}\) is injective, and therefore 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 results 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 \(p\), \(\{ s \in H^0(X, \mathcal{L}) : s(p) = 0 \}\), 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

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

by sections vanishing at \(p\) spans the whole \(\mathfrak{m}_p\mathcal{L}_p / \mathfrak{m}_p^2\mathcal{L}_p\). It is not difficult to verify that the converses also hold. That is, the following holds.

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

Now let us see how these separation conditions are verified via cohomology. First, in the case of (1), consider the closed subvariety \(Z = \{p\} \cup \{q\}\) 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

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

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 becomes surjective and we see that separation of points holds.

Similarly, in the case of (2), considering the first infinitesimal neighborhood \(\operatorname{Spec}(\mathcal{O}_{X,p}/\mathfrak{m}_p^2)\) of the point \(p\) and letting \(\mathcal{I}_p\) 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

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

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

Specifically, if \(\mathcal{L}\) is ample, Kodaira vanishing guarantees \(H^i(X, \omega_X \otimes \mathcal{L}^{\otimes m}) = 0\) (\(i > 0\)). Using appropriate twists and Serre duality, the above \(H^1\) groups also vanish, so we can show that sections of \(\mathcal{L}^{\otimes m}\) satisfy the above separation conditions for sufficiently large \(m\). This is used crucially in the proof of the Kodaira embedding theorem in Proposition 6. Furthermore, the condition that \(\mathcal{L}^{\otimes m}\) is very ample and that the embedding it defines is projectively normal can also be obtained by verifying the surjectivity of the associated multiplication map

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

via Kodaira vanishing. Such vanishing ensures that higher cohomology does not obstruct the generation of sections, allowing us to treat the richness 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 called a Kähler manifold if 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 \(\mathcal{L}\) is called positive if \(\frac{i}{2\pi}\Theta_h\) 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; in particular, \(\mathcal{L}\) is an ample line bundle. Therefore \(X\) is a projective variety.

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

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References

[Har] 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.