Survey-k¨ahler manifolds of positive ricci curvature

Abstract EN

The aim of this paper is to give a short introduction to K¨ahler manifolds of positive Ricci curvature.

It was proved by S.

T.

Yau, as a consequence of his solution of the famous Calabi conjecture, that a compact K¨ahler manifold admits a K¨ahler metric of positive Ricci curvature if and only if its ant canonical line bundle is ”positive”.

By Kodaira’s embedding theorem, a compact K¨ahler manifold of positive Ricci curvature is projective (called a Fano manifold).

Yau and Kodaira’s results opened the door for the investigation of compact K¨ahler manifolds of positive Ricci curvature using both algebra-geometric and differential geometric tools.

The first part of the paper deals with Fano manifolds.

Our approach will be differential geometric.

The picture is less clear in the noncom pact case.

The structure of complete non-compact K¨ahler manifolds of positive Ricci curvature is less understood compared to the compact case, there are some partial results but no general theory.

In the second part of the paper we will review some of those results.

For the sake of completeness, a short introduction to Riemannian geometry is included.