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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2509.20685 |
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| _version_ | 1866915512920834048 |
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| author | Doan, Aleksander Muñoz-Echániz, Juan |
| author_facet | Doan, Aleksander Muñoz-Echániz, Juan |
| contents | We construct Morse homology groups associated with any regular function on a smooth complex algebraic variety, allowing singular and non-compact critical loci. These groups are generated by critical points of a certain large pertubation of the function, built from a normal crossing compactification of the variety. They are canonically isomorphic to the homology of vanishing cycles and -- in the absence of bifurcations at infinity -- recover the hypercohomology of the perverse sheaf of vanishing cycles, studied extensively in singularity theory and enumerative geometry. Our construction arises as a special case of a more general construction of Morse homology of non-compact manifolds that admit a compactification by a manifold with corners. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20685 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Morse complex for the homology of vanishing cycles Doan, Aleksander Muñoz-Echániz, Juan Geometric Topology Algebraic Geometry Differential Geometry We construct Morse homology groups associated with any regular function on a smooth complex algebraic variety, allowing singular and non-compact critical loci. These groups are generated by critical points of a certain large pertubation of the function, built from a normal crossing compactification of the variety. They are canonically isomorphic to the homology of vanishing cycles and -- in the absence of bifurcations at infinity -- recover the hypercohomology of the perverse sheaf of vanishing cycles, studied extensively in singularity theory and enumerative geometry. Our construction arises as a special case of a more general construction of Morse homology of non-compact manifolds that admit a compactification by a manifold with corners. |
| title | A Morse complex for the homology of vanishing cycles |
| topic | Geometric Topology Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2509.20685 |