Perfect correspondences and Chow motives

TL;DR

This thesis introduces perfect correspondences to link derived categories of smooth projective varieties with their Chow motives, advancing understanding of Orlov's conjecture and the invariance of additive invariants.

Contribution

It defines a new category of perfect correspondences and proves that derived equivalences determine both noncommutative and commutative Chow motives up to Tate twists.

Findings

Derived categories determine noncommutative Chow motives.

Derived categories determine Chow motives up to Tate twists.

Additive invariants depend only on the derived category.

Abstract

It is an open conjecture of Orlov that the bounded derived category of coherent sheaves of a smooth projective variety determines its Chow motive with rational coefficients. In this master's thesis we introduce a category of \emph{perfect correspondences}, whose objects are smooth projective varieties and morphisms $X \to Y$ are perfect complexes on $X \times Y$. We show that isomorphism in this category is the same as equivalence of derived categories, and use this to show that the derived category determines the noncommutative Chow motive (in the sense of Tabuada) and, up to Tate twists, the commutative Chow motive with rational coefficients. In particular, all additive invariants like K-theory and Hochschild or cyclic homology depend only on the derived category.