Hirzebruch $χ_{y}$-genus of compact almost Kähler manifold with negative sectional curvature

Abstract: Let ((X,J,ω)) be a closed (2n)-dimensional almost Kähler 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 (χ{y})-genus satisfy the inequality ((-1)^{n-p}χ{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}χ(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 Kähler to the almost Kähler setting under the stated smallness condition.

• The paper extends Gromov's Kähler results to almost Kähler manifolds, establishing new L²-estimates for harmonic forms under small Nijenhuis tensor bounds.
• It proves sharp inequalities for the components of the Hirzebruch χ_y-genus, implying strengthened Euler characteristic bounds related to the Hopf conjecture.
• The analysis leverages advanced Hodge theory techniques to overcome non-integrability challenges inherent in almost Kähler settings.

Hirzebruch $\chi_{y}$-Genus of Compact Almost Kähler Manifolds with Negative Sectional Curvature

Introduction

The paper "Hirzebruch $\chi_{y}$-genus of compact almost Kähler manifold with negative sectional curvature" (2604.27423) examines fundamental topological and analytic properties of closed almost Kähler manifolds admitting negatively curved metrics, focusing on the behavior of the Hirzebruch $\chi_{y}$-genus and its implications for classical conjectures such as the Hopf conjecture. The main contribution is to extend Gromov's result for Kähler manifolds to the almost Kähler setting, under certain explicit smallness conditions on the Nijenhuis tensor, and to derive strong inequalities for the components of the Hirzebruch genus using new $L^2$-estimates for harmonic forms.

Background and Notation

An almost Kähler manifold $(X, J, \omega)$ consists of a symplectic form $\omega$ and a compatible almost complex structure $J$; integrability of $J$ is not assumed. The key analytic challenge in this context lies in the lack of standard Kähler identities due to the potential non-integrability measured by the Nijenhuis tensor $N_J$.

For closed Kähler manifolds, Dolbeault cohomology and Hodge theory provide powerful tools to relate analytic and topological invariants. For almost complex (and in particular, almost Kähler) manifolds, however, Dolbeault theory must be generalized due to the non-vanishing of the commutator terms involving non-integrable structures. Recent advances by Cirici and Wilson permit the development of a Hodge theory suitable for the almost Kähler setting, with harmonic forms corresponding to suitably defined Laplacians involving both $\partial$ and $\chi_{y}$0.

The focus of the paper is the Hirzebruch $\chi_{y}$1-genus, a universal multiplicative genus capturing the Euler characteristic, Todd genus, and signature as specializations. For an almost complex manifold $\chi_{y}$2, the genus is defined as

$\chi_{y}$3

where $\chi_{y}$4 denotes the index of the elliptic operator $\chi_{y}$5 acting on forms of type $\chi_{y}$6.

Analytical Framework and Main Estimates

The primary analytical advance involves establishing sharp $\chi_{y}$7-estimates for harmonic forms on the universal covering of $\chi_{y}$8, leveraging the geometry induced by negative sectional curvature. The central technical result asserts that if the Kähler form is $\chi_{y}$9-bounded (i.e., $\chi_{y}$0 for some bounded $\chi_{y}$1-form $\chi_{y}$2) and $\chi_{y}$3 is sufficiently small, then $\chi_{y}$4-harmonic forms vanish in all bidegrees $\chi_{y}$5 except when $\chi_{y}$6.

This result is achieved without reliance on conventional Kähler identities. Instead, the analysis exploits recent advances in Hodge theory on almost Kähler manifolds and combines integration by parts, commutator identities from [CW20], and delicate control of non-integrable terms depending explicitly on $\chi_{y}$7. The sharp estimates are effective only when $\chi_{y}$8 obeys a uniform bound $\chi_{y}$9 proportional to the lower curvature bound $L^2$0 (where $L^2$1).

Topological Implications and Main Results

The pivotal theorem of the paper asserts that for a closed $L^2$2-dimensional almost Kähler manifold $L^2$3 with negative sectional curvature and sufficiently small $L^2$4, the components of the Hirzebruch $L^2$5-genus satisfy:

$L^2$6

In particular, this yields for the Euler characteristic (the Hopf conjecture):

$L^2$7

This is a strict strengthening compared to the classical Hopf conjecture, which asserts only the sign of the Euler characteristic under negative curvature. The proof leverages $L^2$8-estimates and a refined vanishing theorem for $L^2$9, combined with Atiyah’s $(X, J, \omega)$0-index theorem to relate analytic index on the universal cover to the topological invariants of the manifold.

Corollaries include inequalities for the Taylor coefficients $(X, J, \omega)$1 in shifted expansions of $(X, J, \omega)$2, capturing further integrality and positivity properties of these topological invariants.

Relation to Previous Work and Generalization

Gromov's work [Gro, J. Differential Geom., 1991] established these inequalities under the assumption of a Kähler metric. The extension here to almost Kähler manifolds is not formal, since the absence of the full Kähler identities introduces major analytic obstacles, resolved by the new $(X, J, \omega)$3-techniques developed in this work. Furthermore, the result is not purely curvature-based but depends on both the negative sectional curvature and the smallness of the Nijenhuis tensor, an explicit analytic measure of non-integrability.

This result situates almost Kähler geometry as an intermediate case between symplectic and complex geometry, where topological rigidity of curvature and analytic invariants are tightly coupled only under explicit analytic bounds.

Future Directions and Theoretical Implications

The methodology of this work suggests several avenues for future research:

• Quantification of the Nijenhuis Bound: The explicit dependence of the results on $(X, J, \omega)$4 raises the question of how the critical value of $(X, J, \omega)$5 depends on $(X, J, \omega)$6 and whether the bounds can be further relaxed or made effective for explicit families of almost Kähler manifolds.
• Generalization to Non-compact or Infinite Volume Manifolds: Techniques based on $(X, J, \omega)$7-index theory, as adapted here, could potentially be extended to broader classes of almost complex or symplectic manifolds with controlled noncompactness or nontrivial fundamental groups.
• Applications to Floer- and Symplectic Topology: The topological rigidity phenomena here may interact with other rigidity properties in symplectic topology, e.g., constraints on Lagrangian submanifolds or Fukaya categories under negative curvature assumptions.
• Analytic Aspects in Kähler-Ricci Flow: Understanding the interaction of curvature bounds, almost complex structures, and $(X, J, \omega)$8-Hodge theory could inform flow methods and convergence criteria for canonical metrics.

Conclusion

This work advances the understanding of the interaction between curvature, almost complex structures, and topological invariants on compact symplectic manifolds. By extending the reach of index-theoretic and $(X, J, \omega)$9-analytic techniques to the almost Kähler regime, under precisely quantified integrability bounds, it substantiates and sharpens longstanding conjectures in global differential geometry. The explicit inequalities for all components of the Hirzebruch $\omega$0-genus highlight a previously unattainable level of topological rigidity and establish new benchmarks for future investigations in complex and symplectic geometry.