Derived categories of Fano threefolds

TL;DR

This paper explores the structure of derived categories of coherent sheaves on Fano threefolds with Picard number 1 and uncovers intriguing relations between their derived categories, also relating algebraic cycles to Grothendieck groups.

Contribution

It provides a detailed analysis of derived categories of Fano threefolds and reveals novel relations between the derived categories of different threefolds.

Findings

Identified specific relations between derived categories of different Fano threefolds.

Connected the ring of algebraic cycles to the Grothendieck group of the derived category.

Enhanced understanding of the categorical structure of Fano threefolds.

Abstract

We consider the structure of the derived categories of coherent sheaves on Fano threefolds with Picard number 1 and describe a strange relation between derived categories of different threefolds. In the Appendix we discuss how the ring of algebraic cycles of a smooth projective variety is related to the Grothendieck group of its derived category.