The Kodaira Vanishing Theorem is a fundamental result in the study of compact complex manifolds. It states that higher cohomology groups of certain line bundles are zero. This can be used to conclude, for instance, that some restriction maps on global sections are surjective. It can also be used to show that the dimension of the space of global sections of some line bundles is an Euler characteristic.
The precise statement of Kodaira vanishing is: Let X be a compact complex manifold, and let L be a line bundle on X which is positive, i.e., admits a Hermitian metric with positive definite curvature. Then the cohomology groups H^q(X, K_X \otimes L) are zero for all q > 0. One corollary of this is that a line bundle is positive iff it is ample.
In this talk we will present a proof of Kodaira vanishing via Hodge theory. (No prior knowledge of Hodge theory will be assumed.) The fundamental insight is that positive curvature restricts the behavior of the Laplacian.