Proof of a Conjecture of Batyrev and Nill
  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. 
David Favero and Tyler L. Kelly
math.AG