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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.27423 |
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| _version_ | 1866918528715587584 |
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| author | Huang, Teng Zhang, Pan |
| author_facet | Huang, Teng Zhang, Pan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_27423 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hirzebruch $χ_{y}$-genus of compact almost Kähler manifold with negative sectional curvature Huang, Teng Zhang, Pan Differential Geometry 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. |
| title | Hirzebruch $χ_{y}$-genus of compact almost Kähler manifold with negative sectional curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2604.27423 |