Height of exceptional collections and Hochschild cohomology of quasiphantom categories

TL;DR

This paper introduces the concept of height for exceptional collections in derived categories, linking it to Hochschild cohomology and providing insights into the structure of quasiphantom categories on surfaces of general type.

Contribution

It defines the new invariant 'height' for exceptional collections and relates it to Hochschild cohomology, offering criteria for fullness and applications to quasiphantom categories.

Findings

Orthogonal to an exceptional collection of height h shares Hochschild cohomology with the variety up to degree h-2.

Provides conditions for fullness of exceptional collections based on height and Hochschild cohomology.

Describes the second Hochschild cohomology of quasiphantom categories in certain surfaces of general type.

Abstract

We define the normal Hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective variety $X$ --- a graded vector space which controls the restriction morphism from the Hochschild cohomology of $X$ to the Hochschild cohomology of the orthogonal complement of this admissible subcategory. When the subcategory is generated by an exceptional collection, we define its new invariant (the height) and show that the orthogonal to an exceptional collection of height $h$ in the derived category of a smooth projective variety $X$ has the same Hochschild cohomology as $X$ in degrees up to $h - 2$. We use this to describe the second Hochschild cohomology of quasiphantom categories in the derived categories of some surfaces of general type. We also give necessary and sufficient conditions of fullness of an exceptional collection in terms of its height and of its normal Hochschild cohomology.