Homological projective duality for determinantal varieties

Abstract

International audienceIn this paper we prove Homological Projective Duality for categorical resolutions of several classes of linear determinantal varieties. By this we mean varieties that are cut out by the minors of a given rank of a m x n matrix of linear forms on a given projective space. As applications, we obtain pairs of derived-equivalent Calabi-Yau manifolds, and address a question by A. Bondal asking whether the derived category of any smooth projective variety can be fully faithfully embedded in the derived category of a smooth Fano variety. Moreover we, discuss the relation between rationality and categorical representability in codimension two for determinantal varieties. (C) 2016 Elsevier Inc. All rights reserved

Derived categories of curves as components of Fano manifolds

“…In fact, this conjecture would give a positive answer in dimension 1 to a general question asked by Alexei Bondal some time ago: whether for any algebraic variety X there is Fano variety Y and a fully faithful embedding D(X) → D(Y ) (see [1,6,7] for other results in this direction).…”

Derived Categories of Families of Sextic del Pezzo Surfaces

“…is described in [BBF16]. For this the linear homological projective duality argument [Kuz07, Section 8] is used.…”

All complete intersection varieties are Fano visitors

“…In 2011, Bondal raised the following question (cf. [5]). If the answer to this question is yes for all smooth projective varieties, namely if all smooth projective varieties are Fano visitors, then we can effectively reduce the study of the derived categories of all smooth projective varieties to the Fano case.…”

“…Here is the current state of knowledge for Bondal's question: In [5], Bernardara, Bolognesi and Faenzi use homological projective duality to find a list of Fano visitors, including all smooth plane curves. In [28], Segal and Thomas prove that for a general quintic 3-fold Y , there is a Fano 11-fold X such that there is a fully faithful…”