There are a number of results in geometry which allow to conclude that a certain group vanishes or is bounded under hypotheses on the curvature. For instance, we have:

Theorem 1 If ${M}$ is a compact manifold of positive curvature, then ${H^1(M; \mathbb{R}) = 0}$.

Another such result is the Kodaira vanishing theorem, which enables one to show that certain cohomology groups of an ample line bundle on a smooth projective variety vanish in characteristic zero.

I’ve been trying to gain an understanding of such results, and it seems that there is a common technique in such arguments. The first strategy is to identify the desired cohomology group (e.g. ${H^p(M; \mathbb{R})}$) with the kernel of a Laplacian-type operator, by Hodge theory. The second step is to bound below the relevant Laplacian-type operator. In this post, I’d like to try to explain what’s going on, in a special case. (more…)