# Derived categories and birationality

@article{Lieblich2020DerivedCA, title={Derived categories and birationality}, author={Max Lieblich and Martin C. Olsson}, journal={arXiv: Algebraic Geometry}, year={2020} }

We discuss the question of finding conditions on a derived equivalence between two smooth projective varieties $X$ and $Y$ that imply that $X$ and $Y$ are birational. The types of conditions we consider are in the spirit of finding categorical analogous of classical Torelli theorems. We study, in particular, a notion of strongly filtered derived equivalence and study cases where strongly filtered derived equivalence implies birationality. We also consider an open variant of our main question.

## One Citation

Admissible subcategories of del Pezzo surfaces

- Mathematics
- 2020

We study admissible subcategories of derived categories of coherent sheaves on del Pezzo surfaces and rational elliptic surfaces. Using a relation between admissible subcategories and anticanonical…

## References

SHOWING 1-10 OF 28 REFERENCES

A Stronger Derived Torelli Theorem for K3 Surfaces

- Mathematics
- 2017

In an earlier paper the notion of a filtered derived equivalence was introduced, and it was shown that if two K3 surfaces admit such an equivalence, then they are isomorphic. In this paper we study…

A reconstruction theorem for varieties

- Mathematics
- 2019

We show that varieties of dimension at least 2 over infinite fields are determined as abstract schemes by their Zariski topological spaces together with the rational equivalence relation on the set…

Derived invariants of irregular varieties and Hochschild homology

- Mathematics
- 2014

We study the behavior of cohomological support loci of the canonical bundle under derived equivalence of smooth projective varieties. This is achieved by investigating the derived invariance of a…

DERIVED CATEGORIES OF COHERENT SHEAVES ON ABELIAN VARIETIES AND EQUIVALENCES BETWEEN THEM

- Mathematics
- 1997

We study derived categories of coherent sheaves on Abelian varieties. We give a criterion for the equivalence of the derived categories on two Abelian varieties and describe the autoequivalence group…

Motivic Realizations of Singularity Categories and Vanishing Cycles

- Mathematics
- 2016

In this paper we establish a precise comparison between vanishing cycles and the singularity category of Landau-Ginzburg models over a complete discrete valuation ring. By using noncommutative…

Mukai implies McKay: the McKay correspondence as an equivalence of derived categories

- Mathematics
- 1999

Let G be a finite group of automorphisms of a nonsingular complex threefold M such that the canonical bundle omega_M is locally trivial as a G-sheaf. We prove that the Hilbert scheme Y=GHilb M…

Complex surfaces with equivalent derived categories

- Mathematics, Physics
- 2001

Abstract. We examine the extent to which a smooth minimal complex projective surface X is determined by its derived category of coherent sheaves (DX). To do this we find, for each such surface X, the…

Remarks on Grothendieck's standard conjectures

- Mathematics
- 2010

We show that Grothendieck's standard conjectures are implied by either of two other motivic conjectures: (a) by that of the existence of the motivic t-structure, and (b) by (a weak form of) Suslin's…