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

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. 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 $\mathrm{Bir}({\mathbb{P}}^2,\pm \Omega )$, and that all of these automorphisms are mirror to symplectomorphisms of $M_{\mathrm{univ}}$. We conclude with some applications.