K\"ahler manifolds with almost non-negative curvature

TL;DR

This paper develops methods to analyze the limits of certain non-collapsed K"ahler manifolds with curvature bounds, showing they are homeomorphic to complex manifolds and exploring their complex structures.

Contribution

It constructs solutions to the K"ahler-Ricci flow on non-collapsed manifolds and proves the Gromov-Hausdorff limits are homeomorphic to complex manifolds, advancing understanding of their geometric and complex structures.

Findings

Gromov-Hausdorff limits are homeomorphic to complex manifolds.

Solutions to K"ahler-Ricci flow are constructed under curvature bounds.

Analysis of complex structures on manifolds with nonnegative orthogonal bisectional curvature.

Abstract

In this paper, we construct local and global solutions to the K\"ahler-Ricci flow from a non-collapsed K\"ahler manifold with curvature bounded from below. Combines with the mollification technique of McLeod-Simon-Topping, we show that the Gromov-Hausdorff limit of sequence of complete noncompact non-collapsed K\"ahler manifolds with orthogonal bisectional curvature and Ricci curvature bounded from below is homeomorphic to a complex manifold. We also use it to study the complex structure of complete K\"ahler manifolds with nonnegative orthogonal bisectional curvature, nonnegative Ricci curvature and maximal volume growth.