[Math] Intuition behind the Kodaira Vanishing Theorem

Question

As the question suggests, what is the intuition behind the Kodaira Vanishing Theorem? The Kodaira Vanishing Theorem says that the cohomology groups $H^q(M, L \otimes K_M)$ vanish for $q \ge 1$ when $L$ is a holomoprhic line bundle over a compact complex manifold $M$ and $L$ admits a Hermitian metric whose curvature form is positive definite. Here, $K_M$ denotes the canonical line bundle.

Answer 1

Here are some thoughts. For simplicity I will call a line bundle positive if it admits a metric as you described.

According to Serre's vanishing theorem for any (fixed) coherent sheaf $F$, the sheaf $F\otimes L^m$ has no higher cohomology for $m\gg 0$. So as a first approximation you could consider Kodaira's vanishing theorem as a more precise version of that in the case when $F$ is the canonical line bundle.

It might also help to look at examples.

As Adam points out, the statement is trivial for curves: Using Serre duality the (correct) statement is equivalent to $$ H^q(M, L^{-1}) =0 \text{ for } q<\dim M. $$ So if $\dim M=1$, then this only says that $H^0(M, L^{-1}) =0$ (which is obvious).

One can also look at the cohomology of line bundles on a projective space. In that case the canonical line bundle is just $\mathscr O_{\mathbb P^n}(-n-1)$ and that's exactly so to speak the "breaking point" where the cohomology groups start behaving differently.

As an alternative, one could say that Kodaira's vanishing theorem is a vast generalization of the behaviour of the cohomology of line bundles on projective spaces.

Next, as an exercise, you should do the same comparison for complete intersections. You will find two things:

1. The pattern is similar, the behaviour of cohomology groups changes when you hit the canonical line bundle and what matters whether your line bundle is more or less positive than the canonical.
2. You will discover that the canonical line bundle itself tends to be positive. A simple form of Serre's vanishing says that a high power of a positive line bundle has no positive cohomology. Kodaira's vanishing is again a more precise version of that; measuring by the canonical line bundle it tells how high a power you need. In particular, if the canonical line bundle itself is positive, then Kodaira says that Serre vanishing hold for this particular line bundle already from the second power (and by Serre duality you know that it cannot hold for the first power). Given the importance of the canonical line bundle this is pretty useful.