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

TL;DR

This paper establishes a homological mirror symmetry framework for the pair $(\mathbb{P}^2, \Omega)$, constructing universal objects on both sides and analyzing their automorphisms, linking birational geometry and symplectic symmetries.

Contribution

It constructs universal objects in the homological mirror symmetry setting for $(\mathbb{P}^2, \Omega)$ and relates automorphisms to birational and symplectic transformations.

Findings

Homological mirror symmetry proven for universal objects.

Automorphisms correspond to a subgroup of birational transformations.

Automorphisms are mirror to symplectomorphisms.

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.