Semiorthogonal decomposition for algebraic varieties
A. Bondal, D. Orlov
TL;DR
Bondal and Orlov develop a criterion for full-faithful functors between derived categories and use it to derive semiorthogonal decompositions. They construct a semiorthogonal decomposition for the derived category of the intersection of two quadrics by embedding the derived category of a hyperelliptic curve into the variety and generating the remainder with an exceptional collection. They investigate how $D^b_{coh}$ behaves under birational transformations, proving embeddings for flips and equivalences for flops, and prove a reconstruction theorem showing that varieties with ample canonical or anti-canonical class are determined by their derived category. The work links moduli problems, birational geometry, and categorical reconstruction, positioning the derived category as a robust invariant for algebraic varieties.
Abstract
A criterion for a functor between derived categories of coherent sheaves to be full and faithful is given. A semiorthogonal decomposition for the derived category of coherent sheaves on the intersection of two even dimensional quadrics is obtained. The behaviour of derived categories with respect to birational transformations is investigated. A theorem about reconstruction of a variety from the derived category of coherent sheaves is proved.