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.
Categories: math.AG