D-equivalence and K-equivalence

TL;DR

This paper explores the relationship between D-equivalence and K-equivalence of smooth projective varieties, proposing that these notions may coincide for birationally equivalent varieties, and offers partial insights into this conjecture.

Contribution

It provides a partial answer to whether D-equivalence and K-equivalence coincide for birationally equivalent smooth projective varieties.

Findings

D-equivalence and K-equivalence may coincide for certain varieties

Partial results supporting the conjecture

Framework for comparing derived and birational equivalences

Abstract

Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$. They are called {\it $D$-equivalent} if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories, while {\it $K$-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 at least for birationally equivalent varieties. We shall provide a partial answer to the above problem in this paper.