Dmitrii Pirozhkov - Stably semiorthogonally indecomposable varieties

epiga:7700 - Épijournal de Géométrie Algébrique, March 27, 2023, Volume 7 -
Stably semiorthogonally indecomposable varietiesArticle

Authors: Dmitrii Pirozhkov ORCID

    A triangulated category is said to be indecomposable if it admits no nontrivial semiorthogonal decompositions. We introduce a definition of a noncommutatively stably semiorthogonally indecomposable (NSSI) variety. This propery implies, among other things, that each smooth proper subvariety has indecomposable derived category of coherent sheaves, and that if $Y$ is NSSI, then for any variety $X$ all semiorthogonal decompositions of $X \times Y$ are induced from decompositions of $X$. We prove that any variety whose Albanese morphism is finite is NSSI, and that the total space of a fibration over NSSI base with NSSI fibers is also NSSI. We apply this indecomposability to deduce that there are no phantom subcategories in some varieties, including surfaces $C \times \mathbb{P}^1$, where $C$ is any smooth proper curve of positive genus.

    Volume: Volume 7
    Published on: March 27, 2023
    Accepted on: December 16, 2022
    Submitted on: July 24, 2021
    Keywords: Mathematics - Algebraic Geometry

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