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

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.