D-Equivalence and K-Equivalence

Abstract

Let X and Y be smooth projective varieties over ℂ. They are called D-equivalent if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories, and K-equivalent if they are birationally equivalent and the pull-backs of their canonical divisors to a common resolution coincide. We expect that the two equivalences coincide for birationally equivalent varieties. We shall provide a partial answer to the above problem in this paper.