Dolbeault-Morse-Novikov Cohomology on Complex manifolds and its applications

TL;DR

This paper explores the Dolbeault-Morse-Novikov cohomology on complex manifolds, establishing conditions under which the cohomology and Hirzebruch genus vanish, thus extending classical Riemannian results to complex geometry.

Contribution

It introduces new vanishing theorems for Dolbeault-Morse-Novikov cohomology and Hirzebruch genus related to the existence of certain (0,1)-forms on complex manifolds.

Findings

Vanishing of Dolbeault-Morse-Novikov cohomology when a nonzero parallel (0,1)-form exists.

Vanishing of the Hirzebruch χ_y-genus when a nowhere vanishing (0,1)-form exists.

Hirzebruch χ_y-genus vanishes if and only if the manifold admits a nowhere vanishing real vector field.

Abstract

In this article, we investigate the topological properties of complex manifolds by studying Dolbeault-Morse-Novikov cohomology. By establishing an integral inequality, we obtain two main results: (1) When a closed complex manifold admits a nonzero parallel $(0,1)$-form, the Dolbeault-Morse-Novikov cohomology must be trivial, which implies that the Hirzebruch $\chi_{y}$-genus vanishes. (2) When a closed complex manifold admits a nowhere vanishing $(0,1)$-form, we establish a vanishing theorem for a certain class of twisted Dirac operators, which also forces the Hirzebruch $\chi_{y}$-genus to be zero. In particular, we prove that the Hirzebruch $\chi_{y}$-genus of a closed complex manifold vanishes if and only if the manifold admits a nowhere vanishing real vector field. These results generalize some classical theorems from Riemannian manifolds to the complex setting. As a culminating application, we prove that the Hirzebruch $\chi_{y}$-genus must vanish on closed Gauduchon manifolds admitting positive holomorphic scalar curvature.