Moduli spaces of semiorthogonal decompositions in families

Abstract: For a smooth projective family of schemes we prove that a semiorthogonal decomposition of the bounded derived category of coherent sheaves of a fibre uniquely deforms over an \'etale neighbourhood of the point. We do this using a comparison theorem between semiorthogonal decompositions and decomposition triangles of the structure sheaf of the diagonal. We then apply to the latter a deformation theory for morphisms with a fixed lift of the target, which is developed in the appendix. Using this as a key ingredient we introduce a moduli space which classifies semiorthogonal decompositions of the category of perfect complexes of a smooth projective family. Using Artin's criterion, we show that this is an \'etale algebraic space over the base scheme of the family, which can be non-quasicompact and non-separated. We generalise this to families of geometric noncommutative schemes in the sense of Orlov. We also define a subfunctor classifying nontrivial semiorthogonal decompositions only, and conjecture it is an open and closed subspace.