Derived Categories View on Rationality Problems

Abstract

Abstract. We discuss a relation between the structure of derived categories of smooth projective varieties and their birational properties. We suggest a possible definition of a birational invariant, the derived category analogue of the intermediate Jacobian, and discuss its possible applications to the geometry of prime Fano threefolds and cubic fourfolds.These lectures were prepared for the summer school "Rationality Problems in Algebraic Geometry" organized by CIME-CIRM in Levico Terme in June 2015. I would like to thank Rita Pardini and Pietro Pirola (the organizers of the school) and all the participants for the wonderful atmosphere.

Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

“…The work of many authors now provides evidence for the usefulness of semiorthogonal decompositions in the birational study of complex projective varieties of dimension at most 4. A survey can be found in . At the same time, the relevance of semiorthogonal decompositions to other areas of algebraic and noncommutative geometry has been growing rapidly, as Kuznetsov points out in his address to the 2014 ICM in Seoul.…”

“…A feature of these semiorthogonal decompositions is that the ‐birational class of is determined by the unordered pair of semisimple algebras up to pairwise Morita equivalence. As pointed out by Kuznetsov [, § 3], one of the most tempting ideas in the theory of semiorthogonal decompositions with a view toward rationality is to define, in any dimension and independently of the base field, a categorical analog of the Griffiths component of the intermediate Jacobian of a complex threefold. Such an analog would be the best candidate for a birational invariant.…”

Homological projective duality for determinantal varieties

“…The motivating ideas of question 1.2 can be traced back to the work of Bondal and Orlov, and to their address at the 2002 ICM [11], and to Kuznetsov's remarkable contributions (e.g. [23] or [31]). Notice that a projective space is representable in dimension 0.…”

Gluing semi-orthogonal decompositions

“…We will follow closely the account of psod-s contained in the previous paper of the authors [27]. We refer the reader to [20] for a comprehensive survey of the classical theory of semi-orthogonal decompositions.…”

“…In this paper we prove a result of this type for semi-orthogonal decompositions (sod-s). These are categorified analogues of direct sum decompositions of abelian groups which have long played a key role in algebraic geometry, see [20] for a survey of results. In fact, it is more convenient to work with the slightly more sophisticated concept of preordered semi-orthogonal decompositions (psod-s), where the factors C w of a dg-category C are labeled by elements of a preorder (P, ≤).…”