Seidel's Mirror Map for Abelian Varieties

TL;DR

The paper extends Seidel's mirror map to higher-dimensional abelian varieties by computing the mirror's homogeneous coordinate ring $\mathcal{R}$ from the Fukaya category, with $\mathcal{R} = \bigoplus_{k=0}^\infty \mathrm{Hom}_{D\mathrm{Fuk}(X)}(\psi(\mathcal{O}), \psi(\mathcal{O}(k)))$ and the endomorphism given by tensoring with $\mathcal{O}_Y(1)$ realized by the symplectomorphism $\rho$ encoding large complex structure monodromy. Crucially, only planar holomorphic disks contribute to Fukaya products, enabling explicit computation and linking the product structure to theta-function data on the dual mirror $\tilde{T}$; this yields an isomorphism between $\mathcal{R}$ and the homogeneous coordinate ring of the embedded mirror. Varying $\rho$ yields distinct mirror families: strictly linear $\rho$ recovers commutative mirrors, affine translations produce embeddings into noncommutative projective spaces (via Sklyanin algebras) or quasihomogeneous embeddings, and Kummer quotients demonstrate the method's applicability to abelian quotients. The results connect symplectic disk counts, theta-function algebras, and noncommutative projective geometry, providing concrete mirrors for abelian varieties and illuminating the role of $\rho$ in modulating the mirror geometry.

Abstract

We compute Seidel's mirror map for abelian varieties by constructing the homogeneous coordinate rings from the Fukaya category of the symplectic mirrors. The computations are feasible as only linear holomorphic disks contribute to the Fukaya composition in the case of the planar Lagrangians used. The map depends on a symplectomorphism $ρ$ representing the large complex structure monodromy. For the example of the two-torus, different families of elliptic curves are obtained by using different $ρ$ which are linear in the universal cover. In the case where $ρ$ is merely affine linear in the universal cover, the commutative elliptic curve mirror is embedded in noncommutative projective space. The case of Kummer surfaces is also considered.