Hirzebruch $\chi_{y}$-genus of compact almost K\"{a}hler manifold with negative sectional curvature

TL;DR

This paper proves inequalities for the Hirzebruch \\chi_{y}-genus of certain almost Kähler manifolds with negative sectional curvature, extending classical results and confirming the Hopf conjecture in this context.

Contribution

It extends Gromov's classical results from Kähler to almost Kähler manifolds under a small Nijenhuis tensor condition, using new harmonic form estimates.

Findings

Inequalities for the Hirzebruch \\chi_{y}-genus components are established.

The Hopf conjecture is confirmed for this class of manifolds.

New L^2-estimates and vanishing theorems are developed for harmonic forms.

Abstract

Let \((X,J,\omega)\) be a closed \(2n\)-dimensional almost K\"{a}hler manifold with negative sectional curvature. We prove that if the Nijenhuis tensor of the almost complex structure is sufficiently small, then the components of the Hirzebruch \(\chi_{y}\)-genus satisfy the inequality \((-1)^{n-p}\chi_{p}(X)\geq 1\) for all \(p=0,1,\cdots,n\). In particular, this result implies the Hopf conjecture in this setting, namely that the Euler number satisfies \((-1)^{n}\chi(X)\geq n+1\). The proof is based on new \(L^{2}\)-estimates for harmonic forms on the universal covering, combined with a refined vanishing theorem for the operator \(\bar{\partial}+\bar{\partial}^{*}\) and Atiyah's \(L^{2}\)-index theorem. This work extends the classical result of Gromov [J. Differential Geom., 1991] from the K\"{a}hler to the almost K\"{a}hler setting under the stated smallness condition.