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

TL;DR

This work develops a universal mirror framework for the pair $(\mathbb{P}^2, \Omega)$ within birational geometry. It constructs two universal objects, $U_{\mathrm{univ}}$ (an open Calabi–Yau) and $M_{\mathrm{univ}}$ (an infinite-type Weinstein manifold), and proves homological mirror symmetry in the universal limit, linking birational automorphisms to exact symplectomorphisms. The paper then identifies a discrete subgroup $\mathrm{Bir}_{e}(\mathbb{P}^2, \pm\Omega)$ whose elements act on the A- and B-side universal categories in a way mirrored by symplectomorphisms of $M_{\mathrm{univ}}$, with HMS transporting pushforwards on coherent categories to symplectic actions. A key result is the injective correspondence between $\mathrm{Bir}_{e}(\mathbb{P}^2, \pm\Omega)$ and $\mathrm{Symp}_{e}(M_{\mathrm{univ}})/(\mathrm{Ham}_{c})$, along with explicit constructions in the E- and GL$_2(\mathbb{Z})$-generated birational moves and their symplectic mirrors. The work culminates with concrete applications to automorphisms of open cubic surfaces, illustrating the mirror bridge between birational dynamics and symplectic mapping class groups. Overall, it provides a canonical, universally compatible mirror framework that ties log Calabi–Yau geometry, wrapped Fukaya categories, and birational transformations into a coherent HMS narrative with broad implications for further studies in cluster theory and open CY geometries.

Abstract

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.

Figures (3)

• Figure 3.1: The birational transformation $(x,y) \mapsto (x^{-1}, y^{-1})$ extends to a biholomorphism between two different toric compactifications of $(\mathbb{C}^*)^2$ to $\mathbb{P}^2$s pictured above.
• Figure 3.2: The cluster transformation $E: (x,y) \mapsto (x, y (1+x)^{-1})$ defines a regular map $\mathrm{Bl}_{(-1, \infty)} \mathbb{P}^1 \times \mathbb{P}^1 \to \mathrm{Bl}_{(-1, 0)} \mathbb{F}_1$.
• Figure 3.3: The sequence of moves recording the factorization of the reflection $r_{p_2}$; the rays illustrate the divisors appearing in the relevant compactifications of $U_{C}$. On a Zariski open set, this automorphism takes a point in the cubic to the other point colinear with the intersection point $y_{p_2}$ of the divisors corresponding to the rays $(1,0)$ and $(-1,-1)$ in the above diagram. In order to resolve $r_{p_2}: Y_C \dashrightarrow Y_C$, we have to blow up $p_2$, thus creating a new divisor corresponding to the grey ray in the first diagram; note that this does not change the interior $U_C$ nor the the mirror $M_C$.

Theorems & Definitions (63)

• Theorem 1.1: Theorem \ref{['thm:autisos']}
• Theorem 1.2: Corollary \ref{['cor:universal-exact-hms']} and Theorem \ref{['thm:mirror-map']}
• Corollary 1.3: Corollary \ref{['cor:mirror-symplecto-fixed']}
• Definition 2.2
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• proof
• Definition 2.7
• Lemma 2.8