Angular pair-of-pants decompositions of complex varieties
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910032397860864 |
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| author | Elmaazouz, Yassine Helminck, Paul Alexander |
| author_facet | Elmaazouz, Yassine Helminck, Paul Alexander |
| contents | We define the notion of torically hyperbolic varieties and we construct pair-of-pants decompositions for these in terms of angle sets of essential projective hyperplane complements. This construction generalizes the classical pair-of-pants decomposition for hyperbolic Riemann surfaces. In our first main theorem, we prove that the natural angle map associated to an essential projective hyperplane complement is a homotopy equivalence, extending earlier work of Salvetti and Björner-Ziegler. By a topological argument, we further show that the angle map for a finite Kummer covering of an essential projective hyperplane complement is likewise a homotopy equivalence. We then explain how these local building blocks can be glued along the dual intersection complex of a semistable degeneration. Using the theory of Kato-Nakayama spaces, we prove that the resulting space is homotopy equivalent to the original algebraic variety. We make this explicit for complete intersections in projective space using techniques from tropical geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_14116 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Angular pair-of-pants decompositions of complex varieties Elmaazouz, Yassine Helminck, Paul Alexander Algebraic Geometry Algebraic Topology 14A21, 52C35, 14T20, 55P10, 55P15, 14D06 We define the notion of torically hyperbolic varieties and we construct pair-of-pants decompositions for these in terms of angle sets of essential projective hyperplane complements. This construction generalizes the classical pair-of-pants decomposition for hyperbolic Riemann surfaces. In our first main theorem, we prove that the natural angle map associated to an essential projective hyperplane complement is a homotopy equivalence, extending earlier work of Salvetti and Björner-Ziegler. By a topological argument, we further show that the angle map for a finite Kummer covering of an essential projective hyperplane complement is likewise a homotopy equivalence. We then explain how these local building blocks can be glued along the dual intersection complex of a semistable degeneration. Using the theory of Kato-Nakayama spaces, we prove that the resulting space is homotopy equivalent to the original algebraic variety. We make this explicit for complete intersections in projective space using techniques from tropical geometry. |
| title | Angular pair-of-pants decompositions of complex varieties |
| topic | Algebraic Geometry Algebraic Topology 14A21, 52C35, 14T20, 55P10, 55P15, 14D06 |
| url | https://arxiv.org/abs/2601.14116 |