Heat kernel and local index theorem for open complex manifolds with $\mathbb{C}^{\ast }$-action
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915236583309312 |
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| author | Cheng, Jih-Hsin Hsiao, Chin-Yu Tsai, I-Hsun |
| author_facet | Cheng, Jih-Hsin Hsiao, Chin-Yu Tsai, I-Hsun |
| contents | For a complex manifold $Σ$ with $\mathbb{C}^{\ast }$-action, we define the $m$-th $\mathbb{C}^{\ast }$ Fourier-Dolbeault cohomology group and consider the $m$-index on $Σ$. By applying the method of transversal heat kernel asymptotics, we obtain a local index formula for the $m$-index. We can reinterpret Kawasaki's Hirzebruch-Riemann-Roch formula for a compact complex orbifold with an orbifold holomorphic line bundle by our integral formulas over a (smooth) complex manifold and finitely many complex submanifolds arising from singular strata. We generalize $\mathbb{C}^{\ast }$-action to complex reductive Lie group $G$-action on a compact or noncompact complex manifold. Among others, we study the nonextendability of open group action and the space of all $G$-invariant holomorphic $p$-forms. Finally, in the case of two compatible holomorphic $\mathbb{C}^{\ast }$-actions, a mirror-type isomorphism is found between two linear spaces of holomorphic forms, and the Euler characteristic associated with these spaces can be computed by our $\mathbb{C}^{\ast }$ local index formula on the total space. In the perspective of the equivariant algebraic cobordism theory $Ω_{\ast }^{\mathbb{C}^{\ast }}(Σ),$ a speculative connection is remarked. Possible relevance to the recent development in physics and number theory is briefly mentioned. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11037 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Heat kernel and local index theorem for open complex manifolds with $\mathbb{C}^{\ast }$-action Cheng, Jih-Hsin Hsiao, Chin-Yu Tsai, I-Hsun Differential Geometry Analysis of PDEs Complex Variables 32Q55, 58E40, 58J20, 58J35 For a complex manifold $Σ$ with $\mathbb{C}^{\ast }$-action, we define the $m$-th $\mathbb{C}^{\ast }$ Fourier-Dolbeault cohomology group and consider the $m$-index on $Σ$. By applying the method of transversal heat kernel asymptotics, we obtain a local index formula for the $m$-index. We can reinterpret Kawasaki's Hirzebruch-Riemann-Roch formula for a compact complex orbifold with an orbifold holomorphic line bundle by our integral formulas over a (smooth) complex manifold and finitely many complex submanifolds arising from singular strata. We generalize $\mathbb{C}^{\ast }$-action to complex reductive Lie group $G$-action on a compact or noncompact complex manifold. Among others, we study the nonextendability of open group action and the space of all $G$-invariant holomorphic $p$-forms. Finally, in the case of two compatible holomorphic $\mathbb{C}^{\ast }$-actions, a mirror-type isomorphism is found between two linear spaces of holomorphic forms, and the Euler characteristic associated with these spaces can be computed by our $\mathbb{C}^{\ast }$ local index formula on the total space. In the perspective of the equivariant algebraic cobordism theory $Ω_{\ast }^{\mathbb{C}^{\ast }}(Σ),$ a speculative connection is remarked. Possible relevance to the recent development in physics and number theory is briefly mentioned. |
| title | Heat kernel and local index theorem for open complex manifolds with $\mathbb{C}^{\ast }$-action |
| topic | Differential Geometry Analysis of PDEs Complex Variables 32Q55, 58E40, 58J20, 58J35 |
| url | https://arxiv.org/abs/2412.11037 |