Fourier--Mukai partners and generalized Kummer structures on generalized Kummer surfaces of order $3$

TL;DR

This paper investigates how generalized Kummer surfaces of order 3 are determined by their underlying abelian surfaces and explores the relationship between their geometric structures and Fourier--Mukai partners, revealing conditions for isomorphism.

Contribution

It extends the understanding of the link between generalized Kummer surfaces and their abelian surfaces, especially regarding the determination of the transcendental lattice and Hodge structures.

Findings

For certain divisors, the surface determines the transcendental lattice and Hodge structure.

Fourier--Mukai partners lead to isomorphic generalized Kummer surfaces under specific conditions.

The relationship between the surface's properties and the abelian surface's lattice structures depends on divisibility conditions.

Abstract

A generalized Kummer surface $X$ of order $3$ is the minimal resolution of the quotient of an abelian surface $A$ by an order $3$ symplectic automorphism. We study a generalization of a problem of Shioda for classical Kummer surfaces, which is to understand how much $X$ is determined by $A$ and conversely. The surface $X$ posses a big and nef divisor $L_{X}$ such that $L_{X}^{2}=0$ or $2$ mod $6$. We show that for surfaces with $L_{X}^{2}=6k$ with $k\neq0,6\,mod\,9$, the surface $X$ determines the transcendental lattice $T(A)$ of $A$ and the Hodge structure on $T(A)$. Conversely if $A$ and $B$ are Fourier-Mukai partners (i.e. if the Hodge structures of their transcendental lattices are isomorphic) and $Y$ is the generalized Kummer surface which is the minimal resolution of the quotient of $B$ by an order $3$ symplectic automorphism, we obtain that $X$ and $Y$ are isomorphic. These results are also know to hold for surfaces with $L_{X}^{2}=2\,mod\,6$ from a previous work. When $k=0\text{ or }6\,mod\,9,$ we show that $X$ determines $T(A)$ and its Hodge structure, but the converse does not hold in general.