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

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.