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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2404.13481 |
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| _version_ | 1866909308271198208 |
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| author | Jayasinghe, Gayana |
| author_facet | Jayasinghe, Gayana |
| contents | We construct Witten instanton complexes for Kähler Hamiltonian Morse functions on stratified pseudomanifolds with wedge Kähler metrics satisfying a local conformally totally geodesic condition. We use this to extend Witten's holomorphic Morse inequalities for the $L^2$ cohomology of Dolbeault complexes, deriving versions for Poincaré Hodge polynomials, the spin Dirac and signature complexes for which we prove rigidity results, in particular establishing the rigidity of $L^2$ de Rham cohomology for these circle actions. We study formulas for Rarita Schwinger operators, generalize formulas studied by Witten and Gibbons-Hawking for the equivariant signature and extend formulas used to compute NUT charges of gravitational instantons. We discuss conjectural inequalities extending known Lefschetz-Riemann-Roch formulas for other cohomology theories including those of Baum-Fulton-Quart. This article contains the first extension of Witten's holomorphic Morse inequalities and instanton complexes to singular spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13481 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Holomorphic Witten instanton complexes on stratified pseudomanifolds with Kähler wedge metrics Jayasinghe, Gayana Differential Geometry Complex Variables Spectral Theory 58J20 (Primary) 58A35 (Secondary) We construct Witten instanton complexes for Kähler Hamiltonian Morse functions on stratified pseudomanifolds with wedge Kähler metrics satisfying a local conformally totally geodesic condition. We use this to extend Witten's holomorphic Morse inequalities for the $L^2$ cohomology of Dolbeault complexes, deriving versions for Poincaré Hodge polynomials, the spin Dirac and signature complexes for which we prove rigidity results, in particular establishing the rigidity of $L^2$ de Rham cohomology for these circle actions. We study formulas for Rarita Schwinger operators, generalize formulas studied by Witten and Gibbons-Hawking for the equivariant signature and extend formulas used to compute NUT charges of gravitational instantons. We discuss conjectural inequalities extending known Lefschetz-Riemann-Roch formulas for other cohomology theories including those of Baum-Fulton-Quart. This article contains the first extension of Witten's holomorphic Morse inequalities and instanton complexes to singular spaces. |
| title | Holomorphic Witten instanton complexes on stratified pseudomanifolds with Kähler wedge metrics |
| topic | Differential Geometry Complex Variables Spectral Theory 58J20 (Primary) 58A35 (Secondary) |
| url | https://arxiv.org/abs/2404.13481 |