Holomorphic one-forms, fibrations over the circle, and characteristic numbers of Kähler manifolds

D Kotschick

doi:10.1017/s0305004121000128

Abstract

AbstractWe prove that the only relation imposed on the Hodge and Chern numbers of a compact Kähler manifold by the existence of a nowhere zero holomorphic one-form is the vanishing of the Hirzebruch genus. We also treat the analogous problem for nowhere zero closed one-forms on smooth manifolds.

Highlights

By the Poincaré–Hopf theorem, a closed smooth manifold admits a vector field without zeros if and only if its Euler characteristic vanishes
Our proof of Theorem 1 will show that the conclusion about the vanishing of the Hirzebruch genus is true for all compact Kähler manifolds that admit a smooth fibration over the circle, and that no other relations are imposed on the Hodge and Chern numbers by the existence of such a fibration
As we mentioned in the introduction, the existence of a nowhere zero holomorphic vector field forces all the Chern numbers to vanish, by a result of Bott [4]

Summary

Introduction

By the Poincaré–Hopf theorem, a closed smooth manifold admits a vector field without zeros if and only if its Euler characteristic vanishes. By Tischler’s theorem [18] the existence of a closed one-form without zeros is equivalent to the existence of a smooth fibration over the circle Such a fibration implies the vanishing of the signature. The main purpose of this paper is to prove Kähler analogs of the above results It is a consequence of Bott’s localisation formula [4] that on a compact complex manifold with a holomorphic vector field without zeros all Chern numbers vanish.

KOTSCHICK

Holomorphic one-forms without zeros on Kähler manifolds

Discussion