Characteristic numbers, Jiang subgroup and non-positive curvature

TL;DR

This paper establishes conditions involving the Jiang subgroup that lead to the vanishing of certain characteristic numbers on compact manifolds, especially in the context of non-positive curvature and Kähler geometry.

Contribution

It refines Farrell's idea to identify conditions under which signature and Hirzebruch's $ ext{chi}_y$-genus vanish, and explores implications for Chern numbers in non-positively curved Kähler manifolds.

Findings

$ ext{chi}_y$-genus vanishes when the fundamental group's center is non-trivial.

All Chern numbers vanish for complex dimension ≤ 4 in certain non-positively curved Kähler manifolds.

Provides partial answers to Farrell's question and supports a conjecture by the author and Zheng.

Abstract

By refining an idea of Farrell, we present a sufficient condition in terms of the Jiang subgroup for the vanishing of signature and Hirzebruch's $\chi_y$-genus on compact smooth and K\"{a}hler manifolds respectively. Along this line we show that the $\chi_y$-genus of a non-positively curved compact K\"{a}hler manifold vanishes when the center of its fundamental group is non-trivial, which partially answers a question of Farrell. Moreover, in the latter case all the Chern numbers vanish whenever its complex dimension is no more than $4$, which also provides some evidence to a conjecture proposed by the author and Zheng.