A universal mirror to $(\mathbb{P}^2, Ω)$ as a birational object
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912650105978880 |
|---|---|
| author | Keating, Ailsa Ward, Abigail |
| author_facet | Keating, Ailsa Ward, Abigail |
| contents | We study homological mirror symmetry for $(\mathbb{P}^2, Ω)$ viewed as an object of birational geometry, with $Ω$ the standard meromorphic volume form. First, we construct universal objects on the two sides of mirror symmetry, focusing on the exact symplectic setting: a smooth complex scheme $U_\mathrm{univ}$ and a Weinstein manifold $M_\mathrm{univ}$, both of infinite type; and we prove homological mirror symmetry for them. Second, we consider autoequivalences. We prove that automorphisms of $U_\mathrm{univ}$ are given by a natural discrete subgroup of $\operatorname{Bir} (\mathbb{P}^2, \pm Ω)$; and that all of these automorphisms are mirror to symplectomorphisms of $M_\mathrm{univ}$. We conclude with some applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03764 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A universal mirror to $(\mathbb{P}^2, Ω)$ as a birational object Keating, Ailsa Ward, Abigail Symplectic Geometry Algebraic Geometry We study homological mirror symmetry for $(\mathbb{P}^2, Ω)$ viewed as an object of birational geometry, with $Ω$ the standard meromorphic volume form. First, we construct universal objects on the two sides of mirror symmetry, focusing on the exact symplectic setting: a smooth complex scheme $U_\mathrm{univ}$ and a Weinstein manifold $M_\mathrm{univ}$, both of infinite type; and we prove homological mirror symmetry for them. Second, we consider autoequivalences. We prove that automorphisms of $U_\mathrm{univ}$ are given by a natural discrete subgroup of $\operatorname{Bir} (\mathbb{P}^2, \pm Ω)$; and that all of these automorphisms are mirror to symplectomorphisms of $M_\mathrm{univ}$. We conclude with some applications. |
| title | A universal mirror to $(\mathbb{P}^2, Ω)$ as a birational object |
| topic | Symplectic Geometry Algebraic Geometry |
| url | https://arxiv.org/abs/2408.03764 |