Proof of a Conjecture of Batyrev and Nill

TL;DR

This paper proves a conjecture by Batyrev and Nill by establishing derived category equivalences in the Batyrev-Borisov mirror symmetry construction, using singularity categories and GIT variations.

Contribution

It provides the first proof of the Batyrev-Nill conjecture, linking mirror symmetry and derived categories through novel categorical techniques.

Findings

Derived categories are equivalent for various mirrors in Batyrev-Borisov construction.

Confirmed a conjecture of Batyrev and Nill regarding mirror symmetry.

Introduced methods involving singularity categories and GIT variations.

Abstract

We prove equivalences of derived categories for the various mirrors in the Batyrev-Borisov construction. In particular, we obtain a positive answer to a conjecture of Batyrev and Nill. The proof involves passing to an associated category of singularities and toric variation of geometric invariant theory quotients.