Categorical Plücker Formula and Homological Projective Duality

Abstract

Homological Projective duality (HP-duality) theory, introduced by Kuznetsov [42], is one of the most powerful frameworks in the homological study of algebraic geometry. The main result (HP-duality theorem) of the theory gives complete descriptions of bounded derived categories of coherent sheaves of (dual) linear sections of HP-dual varieties. We show the theorem also holds for more general intersections beyond linear sections. More explicitly, for a given HP-dual pair (X, Y ), then analogue of HP-duality theorem holds for their intersections with another HP-dual pair (S, T ), provided that they intersect properly. We also prove a relative version of our main result. Taking (S, T ) to be dual linear subspaces (resp. subbundles), our method provides a more direct proof of the original (relative) HPduality theorem.

Blowing up linear categories, refinements, and homological projective duality with base locus

“…Application to homological projective geometry. The work in this paper fits into the framework of homological projective geometry (see [JLX17,KP18,JL18] for more details) as follows. Denote by Lef /P(V ) the category of smooth proper P(V )-linear Lefschetz categories, and fix any linear system (i.e.…”

“…The mutation functors allow us to start with a semiorthogonal decomposition to obtain a whole sequence of new semiorthogonal decompositions. The readers are referred to [H06] and [BO] for definitions and properties of derived categories of algebraic varieties and semiorthogonal decompositions, and to [B, BK] or reviews in [K07,JLX17] for more about mutation functors.…”

“…Lefschetz varieties and categories. Lefschetz decompositions introduced by Kuznetsov in [K07] (see also [K08,JLX17,P18,KP18]) play an essential role in HPD theory. A Lefschetz decomposition is a special kind of semiorthogonal decomposition which behaves well under a given autoequivalence T .…”

“…Homological projective duality (HPD) introduced by Kuznetsov [K07], has been a very fruitful theory to produce interesting semiorthogonal decompositions of derived categories of algebraic varieties, and also to relate derived categories of different varieties, see [T15] and [K14] for nice surveys, or [JLX17] for a review and references therein. An important question is how to produce examples of HPD, and one useful strategy would be to produce new HPDs from existing ones.…”

“…The HPD is a duality relation: namely A is also the HPD of A ♮ . See [K07,T15,JLX17] for more about HPD.…”