Moduli spaces of semiorthogonal decompositions in families

TL;DR

The paper constructs a moduli framework for V-linear semiorthogonal decompositions in smooth, proper families $f:\mathcal X o U$, encoding decompositions of the fibered category $ ext{Perf}\mathcal X_V$ and proving that the resulting moduli $ ext{SOD}_f^ ext{ell}$ is a nonempty étale algebraic space over $U$ when $U$ is excellent. It develops a parallel viewpoint via decompositions of the diagonal (DEC) using Fourier–Mukai kernels, and proves an equivalence between SODs and DECs; this equivalence underpins two proofs that the SOD functor is limit-preserving. The deformation theory shows SODs deform unobstructedly over infinitesimal thickenings and étale neighborhoods, leading to an Artin-style construction of the global moduli space. The work also explores amplifications to subcategories, nontrivial SOD loci, and group actions by mutations and autoequivalences, and discusses detailed examples (notably cubic-surface families) to illustrate geometric and categorical phenomena, including potential open questions about properness and openness of nontrivial loci. Overall, the results provide a broad, versatile moduli-theoretic framework connecting SODs with classical geometry and noncommutative geometry in families, with conjectural extensions to dg-categorical contexts and refined invariants across deformations.

Abstract

To a smooth and proper morphism $\mathcal{X}\to U$ with quasicompact semiseparated target we associate a sheaf in the étale topology, which takes an affine $U$-scheme $V$ to the set of $V$-linear semiorthogonal decompositions (of fixed length) of the category $\operatorname{Perf}\mathcal{X}_V$. We use Artin's criterion to prove that, when $U$ is excellent, this is in fact an algebraic space which is moreover étale (though in general non-quasicompact and non-separated) over $U$. We moreover generalise the construction of the sheaf to families of geometric noncommutative schemes in the sense of Orlov. We also define a subfunctor classifying nontrivial semiorthogonal decompositions, and conjecture it is an open and closed subspace. Along the way, we prove that for a smooth and proper family of schemes, a semiorthogonal decomposition of the bounded derived category of coherent sheaves of a fibre uniquely deforms over an étale neighbourhood of the point.

Theorems & Definitions (146)

• remark 1.2
• example 1.3
• lemma 2.1
• proof
• lemma 2.2: Derived Nakayama--Azumaya--Krull, local version
• proof
• lemma 2.3: Derived Nakayama--Azumaya--Krull, global version
• proof
• definition 2.4
• lemma 2.5: MR2801403