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arXiv · 2408.03764

A universal mirror to $(\mathbb{P}^2, \Omega)$ as a birational object

Ailsa Keating·Abigail Ward
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Abstract

We study homological mirror symmetry for $(\mathbb{P}^2, \Omega)$ viewed as an object of birational geometry, with $\Omega$ 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 $\operatorname{Bir} (\mathbb{P}^2, \pm \Omega)$; and that all of these automorphisms are mirror to symplectomorphisms of $M_\mathrm{univ}$. We conclude with some applications.

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Ailsa Keating, Abigail Ward. 2024-08-07. A universal mirror to $(\mathbb{P}^2, \Omega)$ as a birational object. https://arxiv.org/abs/2408.03764

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