Homological mirror symmetry for the genus two curve

TL;DR

This paper proves a version of Katzarkov's generalized mirror symmetry conjecture for a genus two curve, linking its Fukaya category to Landau-Ginzburg branes on a singular surface, advancing understanding of mirror symmetry for complex varieties.

Contribution

It establishes the first concrete example of Katzarkov's conjecture for a genus two curve, connecting symplectic and algebraic categories in a new setting.

Findings

Proves a version of Katzarkov's mirror symmetry conjecture for genus two curves.

Links the Fukaya category of the curve to Landau-Ginzburg branes on a singular surface.

Provides a new example of mirror symmetry beyond Calabi-Yau varieties.

Abstract

Katzarkov has proposed a generalization of Kontsevich's mirror symmetry conjecture, covering some varieties of general type. We prove a version of this conjecture in the simplest example, relating the Fukaya category of a genus two curve to the category of Landau-Ginzburg branes on a certain singular rational surface.