$S^1$-equivariant Index theorems and Morse inequalities on complex manifolds with boundary
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| Format: | Preprint |
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2017
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| _version_ | 1866913293020430336 |
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| author | Hsiao, Chin-Yu Huang, Rung-Tzung Li, Xiaoshan Shao, Guokuan |
| author_facet | Hsiao, Chin-Yu Huang, Rung-Tzung Li, Xiaoshan Shao, Guokuan |
| contents | Let $M$ be a complex manifold of dimension $n$ with smooth connected boundary $X$. Assume that $\overline M$ admits a holomorphic $S^1$-action preserving the boundary $X$ and the $S^1$-action is transversal on $X$. We show that the $\overline\partial$-Neumann Laplacian on $M$ is transversally elliptic and as a consequence, the $m$-th Fourier component of the $q$-th Dolbeault cohomology group $H^q_m(\overline M)$ is finite dimensional, for every $m\in\mathbb Z$ and every $q=0,1,\ldots,n$. This enables us to define $\sum^{n}_{j=0}(-1)^j{\rm dim\,}H^q_m(\overline M)$ the $m$-th Fourier component of the Euler characteristic on $M$ and to study large $m$-behavior of $H^q_m(\overline M)$. In this paper, we establish an index formula for $\sum^{n}_{j=0}(-1)^j{\rm dim\,}H^q_m(\overline M)$ and Morse inequalities for $H^q_m(\overline M)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1711_05537 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | $S^1$-equivariant Index theorems and Morse inequalities on complex manifolds with boundary Hsiao, Chin-Yu Huang, Rung-Tzung Li, Xiaoshan Shao, Guokuan Complex Variables Analysis of PDEs Differential Geometry Let $M$ be a complex manifold of dimension $n$ with smooth connected boundary $X$. Assume that $\overline M$ admits a holomorphic $S^1$-action preserving the boundary $X$ and the $S^1$-action is transversal on $X$. We show that the $\overline\partial$-Neumann Laplacian on $M$ is transversally elliptic and as a consequence, the $m$-th Fourier component of the $q$-th Dolbeault cohomology group $H^q_m(\overline M)$ is finite dimensional, for every $m\in\mathbb Z$ and every $q=0,1,\ldots,n$. This enables us to define $\sum^{n}_{j=0}(-1)^j{\rm dim\,}H^q_m(\overline M)$ the $m$-th Fourier component of the Euler characteristic on $M$ and to study large $m$-behavior of $H^q_m(\overline M)$. In this paper, we establish an index formula for $\sum^{n}_{j=0}(-1)^j{\rm dim\,}H^q_m(\overline M)$ and Morse inequalities for $H^q_m(\overline M)$. |
| title | $S^1$-equivariant Index theorems and Morse inequalities on complex manifolds with boundary |
| topic | Complex Variables Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/1711.05537 |