The Geometric P=W conjecture and Thurston's compactification
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908806262292480 |
|---|---|
| author | Ayilliath-Kutteri, Ashwin Farajzadeh-Tehrani, Mohammad Frohman, Charles |
| author_facet | Ayilliath-Kutteri, Ashwin Farajzadeh-Tehrani, Mohammad Frohman, Charles |
| contents | In this paper, we use new results together with established facts about Thurston's compactification of Teichmüller space to address the geometric P=W conjecture for $\mathrm{SL}(2,\mathbb{C})$, which concerns projective compactifications of character varieties of closed surfaces. In particular, we construct a projective compactification of the $\mathrm{SL}(2,\mathbb{C})$-character variety of any closed surface of genus $g>1$, in which the boundary divisors are toric varieties and the dual intersection complex is a sphere. A main technical step, of independent interest, is the derivation of an explicit formula for a well-known embedding of the set of isotopy classes of multicurves on a closed surface of genus $g$ into $\mathbb{N}^{9g-9}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07211 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Geometric P=W conjecture and Thurston's compactification Ayilliath-Kutteri, Ashwin Farajzadeh-Tehrani, Mohammad Frohman, Charles Geometric Topology Algebraic Geometry Quantum Algebra 14M35, 57K31 In this paper, we use new results together with established facts about Thurston's compactification of Teichmüller space to address the geometric P=W conjecture for $\mathrm{SL}(2,\mathbb{C})$, which concerns projective compactifications of character varieties of closed surfaces. In particular, we construct a projective compactification of the $\mathrm{SL}(2,\mathbb{C})$-character variety of any closed surface of genus $g>1$, in which the boundary divisors are toric varieties and the dual intersection complex is a sphere. A main technical step, of independent interest, is the derivation of an explicit formula for a well-known embedding of the set of isotopy classes of multicurves on a closed surface of genus $g$ into $\mathbb{N}^{9g-9}$. |
| title | The Geometric P=W conjecture and Thurston's compactification |
| topic | Geometric Topology Algebraic Geometry Quantum Algebra 14M35, 57K31 |
| url | https://arxiv.org/abs/2507.07211 |