Derived categories of hearts on Kuznetsov components

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 $\mathrm{Ku}(X)$ 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 $\mathrm{Ku}(X)\stackrel{\sim }{\to }\mathrm{Ku}(X^{\prime })$ are of Fourier--Mukai type when $X$, $X^{\prime }$ belong to these classes of varieties, as predicted by a conjecture of Kuznetsov.

21 pages