Grauert–Riemenschneider vanishing theorem

In mathematics, the Grauert–Riemenschneider vanishing theorem is an extension of the Kodaira vanishing theorem on the vanishing of higher cohomology groups of coherent sheaves on a compact complex manifold, due to Grauert and Riemenschneider (1970).

Grauert–Riemenschneider conjecture

The Grauert–Riemenschneider conjecture is a conjecture related to the Grauert–Riemenschneider vanishing theorem:

Grauert & Riemenschneider (1970a); Let M be an n-dimensional compact complex manifold. M is Moishezon if and only if there exists a smooth Hermitian line bundle L over M whose curvature form which is semi-positive everywhere and positive on an open dense set.

This conjecture was proved by Siu (1985) using the Riemann–Roch type theorem (Hirzebruch–Riemann–Roch theorem) and by Demailly (1985) using Morse theory.

Note

  1. ↑ (Siu 1985)

References