Derived categories and Kummer varieties

Arithmetic on abelian and Kummer varieties

A Kummer variety is obtained as the quotient of an abelian variety by the automorphism (−1) acting on it. Kummer varieties can be seen as a higher dimensional generalisation of the xcoordinate representation of a point of an elliptic curve given by its Weierstrass model. Although there is no group law on the set of points of a Kummer variety, the multiplication of a point by a scalar still make...

Groups of Autoequivalences of Derived Categories of Smooth Projective Varieties

Derived Categories and Tilting

We review the basic definitions of derived categories and derived functors. We illustrate them on simple but non trivial examples. Then we explain Happel’s theorem which states that each tilting triple yields an equivalence between derived categories. We establish its link with Rickard’s theorem which characterizes derived equivalent algebras. We then examine invariants under derived equivalenc...

Derived Categories , Derived Equivalences and Representation Theory

Deenition: A derived category ... is when you take complexes seriously! (L.L. Scott Sc]) The aim of this chapter is to give a fairly elementary introduction to the (not very elementary) subject of derived categories and equivalences. Especially, we emphasize the applications of derived equivalences in representation theory of groups and algebras in order to illustrate the importance and usefuln...

On equivalences of derived categories of coherent sheaves on abelian varieties

It was proved in [Po1] that if the varieties A × Â and B × B̂ are symplectically isomorphic, where A and B are abelian varieties over algebraically closed field, then the derived categories of coherent sheaves D coh(A) and D b coh(B) are equivalent. The aim of this paper is to give the proof for the inverse statement (see Th.2.7 ), i.e. if the derived categories D coh(A) and D b coh(B) are equiv...