Estimation and Vanishing Results for Hodge numbers

TL;DR

This paper establishes conditions under which compact K"ahler manifolds have cohomology rings similar to complex projective space, through eigenvalue estimates of the K"ahler curvature operator and vanishing theorems for Hodge numbers.

Contribution

It introduces new eigenvalue-based criteria for cohomology ring isomorphism and extends Tachibana's theorem to K"ahler manifolds.

Findings

Cohomology ring of K"ahler manifolds matches that of complex projective space under certain eigenvalue conditions.

New vanishing theorems for individual Hodge numbers are proved.

An analogue of Tachibana's theorem is established for K"ahler manifolds.

Abstract

We show that compact K\"ahler manifolds have the rational cohomology ring of complex projective space provided a weighted sum of the lowest three eigenvalues of the K\"ahler curvature operator is positive. This follows from a more general vanishing and estimation theorem for the individual Hodge numbers. We also prove an analogue of Tachibana's theorem for K\"ahler manifolds.