Birational geometry and derived categories

TL;DR

This paper surveys recent advances in the DK hypothesis, linking birational geometry of smooth projective varieties with their derived categories, proposing that K-equivalence corresponds to derived category equivalence.

Contribution

It provides an overview of recent progress on the DK hypothesis, clarifying the relationship between birational geometry and derived categories.

Findings

K-equivalence implies derived category equivalence

K-inequality implies fully faithful embedding of derived categories

Progress on connecting birational geometry with categorical equivalences

Abstract

This paper is based on a talk at a conference "JDG 2017: Conference on Geometry and Topology". We survey recent progress on the DK hypothesis connecting the birational geometry and the derived categories stating that the K-equivalence of smooth projective varieties should correspond to the equivalence of their derived categories, and the K-inequality to the fully faithful embedding.