Semiorthogonal Decompositions in Algebraic Families

Definitions and mathematical statement

Let $\mathcal D$ be a triangulated category, with shift functor $[1]$. A finite semiorthogonal decomposition \[ \mathcal D=\langle\mathcal A_1,\ldots,\mathcal A_n\rangle, \qquad n\geq 1, \] consists of full triangulated subcategories that generate $\mathcal D$ as a triangulated category and satisfy \[ \operatorname{Hom}_{\mathcal D}(B,A[k])=0 \quad\text{for every }j>i,\ A\in\mathcal A_i,\ B\in\mathcal A_j,\ k\in{\mathbb{Z}}. \] A full triangulated subcategory is called admissible if its inclusion has both a left adjoint and a right adjoint. Whether admissibility is required in the intended decompositions is not specified by the source record.

Source question. How do semiorthogonal decompositions of derived categories behave in algebraic families?

This is a broad research question rather than a specified universally quantified conjecture. The supplied statement does not identify the objects parametrized by a family, the ground field, the base, hypotheses on the family, or the version of the derived category. In particular, it does not choose between bounded derived categories of coherent sheaves and categories of perfect complexes. It also does not specify the intended meaning of behavior in families, such as extension of a decomposition from a fiber, specialization, compatibility with base change, or a moduli description. These are possible interpretations, not additional assertions of the source. Consequently, no precise extension, specialization, or classification assertion can be extracted without further hypotheses and a specified target.

Short English statement

The question asks how semiorthogonal decompositions vary in algebraic families. Such a decomposition expresses a triangulated category through ordered subcategories with vanishing morphisms in one direction. The source leaves both the class of families and the desired behavior unspecified, so this draft preserves the research question and records the missing specifications.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30004468$, source problem code catalogue entry, title "Semiorthogonal Decompositions in Algebraic Families." The supplied record belongs to the Oberwolfach Reports Open Problems collection; no individual proposer is named.
• Supplied source citation: Algebraic Geometry: Moduli Spaces, Birational Geometry and Derived Aspects (2021), Oberwolfach Reports, pp. 977--1021, DOI 10.4171/owr/2020/19. Original OWR report: https://doi.org/10.4171/owr/2020/19. The record separately gives proposed year 2020.
• Supplied literature reference: Bastianelli--Belmans--Okawa--Ricolfi, https://doi.org/10.1016/j.geomphys.2025.105600. The record supplies no title or precise result for this reference.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the question open and reports deformation and moduli results for semiorthogonal decompositions in families, while stating that these do not settle the broad question for arbitrary families. This is supplied dated curation, not an independently verified assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.