Holomorphic Euler number of K$\ddot{a}$hler manifolds with almost nonnegative Ricci curvature

TL;DR

This paper proves that compact Kähler manifolds with almost nonnegative Ricci curvature and nonzero first Betti number have a vanishing holomorphic Euler number, introducing a new obstruction for such geometric structures.

Contribution

It establishes a vanishing theorem for the holomorphic Euler number under specific curvature and topological conditions, advancing understanding of Kähler geometry.

Findings

Holomorphic Euler number vanishes for the specified manifolds

Introduces a new obstruction for Kähler manifolds with almost nonnegative Ricci curvature

Proves a vanishing theorem for Dolbeault-Morse-Novikov cohomology

Abstract

Let $M^n$ be a compact K$\ddot{a}$hler manifold with almost nonnegative Ricci curvature and nonzero first Betti number. We show that the holomorphic Euler number of $M^n$ vanishes, which gives a new obstruction for compact complex manifolds admitting K$\ddot{a}$hler metrics with almost nonnegative Ricci curvature. A crucial step in the proof is to show a vanishing theorem of Dolbeault-Morse-Novikov cohomology.