Derived categories of hearts on Kuznetsov components

TL;DR

This paper establishes a criterion ensuring certain subcategories of derived categories are equivalent to derived categories of hearts, and applies it to Kuznetsov components of specific algebraic varieties, confirming their unique enhancements and Fourier--Mukai equivalences.

Contribution

It provides a general criterion for equivalences to hearts in derived categories and applies it to Kuznetsov components, confirming their unique dg enhancements and Fourier--Mukai nature.

Findings

Kuznetsov components have strongly unique dg enhancements.

Exact equivalences between these components are of Fourier--Mukai type.

The criterion applies to cubic fourfolds, Gushel–Mukai varieties, and quartic double solids.

Abstract

We prove a general criterion which guarantees that an admissible subcategory $\mathcal{K}$ of the derived category of an abelian category is equivalent to the bounded derived category of the heart of a bounded t-structure. As a consequence, we show that $\mathcal{K}$ has a strongly unique dg enhancement, applying the recent results of Canonaco, Neeman and Stellari. We apply this criterion to the Kuznetsov component $\mathop{\mathcal{K}u}(X)$ when $X$ is a cubic fourfold, a Gushel--Mukai variety or a quartic double solid. In particular, we obtain that these Kuznetsov components have strongly unique dg enhancement and that exact equivalences of the form $\mathop{\mathcal{K}u}(X) \xrightarrow{\sim} \mathop{\mathcal{K}u}(X')$ are of Fourier--Mukai type when $X$, $X'$ belong to these classes of varieties, as predicted by a conjecture of Kuznetsov.