Serre functors and dimensions of residual categories

Abstract

We describe in terms of spherical twists the Serre functors of many interesting semiorthogonal components, called residual categories, of the derived categories of projective varieties. In particular, we show the residual categories of Fano complete intersections are fractional Calabi-Yau up to a power of an explicit spherical twist. As applications, we compute the Serre dimensions of residual categories of Fano complete intersections, thereby proving a corrected version of a conjecture of Katzarkov and Kontsevich, and deduce the nonexistence of Serre invariant stability conditions when the degrees of the complete intersection do not all coincide.

Categorical Torelli theorems: results and open problems

“…Remark 2.13. A list of other prime Fano threefolds with Kuznetsov component which is a fractional Calabi-Yau category can be deduced from [73] (see Section 2.4 therein) and [79]. One interesting example is provided by quartic threefolds, i.e.…”

“…One could first wonder whether an admissible subcategory D of D b (X) admits Serre-invariant stability conditions. This has been recently proved to be false in [79], using the notion of Serre dimension, in the case of the Kuznetsov component of almost all Fano complete intersections of codimension ≥ 2. Anyway, we could focus on prime Fano threefolds and consider the following less general question: Consider first the slope stability σ H = (Coh(X), Z H ).…”

On Gromov-Yomdin type theorems and a categorical interpretation of holomorphicity

“…Let Aut(D) be the group of autoequivalences of D and assume that there is a group homomorphism where [29,Section 6.3]. Then all our statements can be generalized by replacing [Φ] Γ by a.…”

“…In general, for any increasing function f : R → R such that f (φ + 1) = f (φ), the proof of [29,Proposition 6.17] shows that…”

Serre-invariant stability conditions and Ulrich bundles on cubic threefolds

“…In the upcoming paper [PR21] the same result is proved for the Kuznetsov component of a Gushel-Mukai threefold. On the other hand, in the recent paper [KP21] the authors show the non-existence of Serre-invariant stability conditions on Kuznetsov components of almost all Fano complete intersections of codimension ≥ 2.…”