Categorical representability and intermediate Jacobians of Fano threefolds

Abstract

Abstract. We define, basing upon semiorthogonal decompositions of D b (X), categorical representability of a projective variety X and describe its relation with classical representabilities of the Chow ring. For complex threefolds satisfying both classical and categorical representability assumptions, we reconstruct the intermediate Jacobian from the semiorthogonal decomposition. We discuss finally how categorical representability can give useful information on the birational properties of X by providing examples and stating open questions.

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

“…Using semiorthogonal decompositions, one can define a notion of categorical representability for a dg‐enhanced triangulated category. In the case of smooth projective varieties, this is inspired by the classical notions of representability of cycles, see . Definition A dg‐enhanced ‐linear triangulated category is representable in dimension if it admits a semiorthogonal decomposition and for each there exists a smooth projective connected ‐variety with , such that is equivalent to an admissible subcategory of .…”

“…Using semiorthogonal decompositions, one can define a notion of categorical representability for a dg-enhanced triangulated category. In the case of smooth projective varieties, this is inspired by the classical notions of representability of cycles, see [22]. Definition 1.14 [22].…”

“…In the case of smooth projective varieties, this is inspired by the classical notions of representability of cycles, see [22]. Definition 1.14 [22]. A dg-enhanced k-linear triangulated category T is representable in dimension m if it admits a semiorthogonal decomposition…”

“…Definition 1.15 [22]. Let X be a projective k-variety of dimension n. We say that X is categorically representable in dimension m (or equivalently in codimension n − m) if there exists a categorical resolution of singularities of D b (X) that is representable in dimension m.…”

“…In this context, based on the classical notion of representability for Chow groups and motives, Bolognesi and the second author proposed the notion of categorical representability (see Definition ), and formulated the following question. Question Is a rational variety always categorically representable in codimension 2?…”