nLab Kodaira vanishing theorem

Idea

The Kodaira vanishing theorem for complex geometry says that if $X$ is a Kähler manifold and $L$ a holomorphic line bundle on $X$ which is positive, then the abelian sheaf cohomology of $X$ with coefficients in the sheaf of sections of the tensor product

$$\Omega^{n,0}_X(L) \simeq L \otimes \Omega^{n,0}_X$$

with the canonical line bundle $\Omega^{n,0}_X$ is concentrated in degree 0:

$$(L \; positive) \;\;\Rightarrow\;\; H^{\bullet \geq 1}(X, \Omega^{n,0}_X(L)) = 0 \,.$$

Related concepts

• Kodaira-Spencer theory

• geometric quantization – Dolbeault quantization

References

The statement is due to Kunihiko Kodaira.

Lecture notes include

• Hélène Esnault, Eckart Viehweg, Lectures on vanishing theorems (1992) DMV Seminar 20 (pdf)