Geodesic
Stably semiorthogonally indecomposable varieties
Épijournal de Géométrie Algébrique, Tome 7 (2023)

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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. 
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     author = {Pirozhkov, Dmitrii},
     title = {Stably semiorthogonally indecomposable varieties},
     journal = {\'Epijournal de G\'eom\'etrie Alg\'ebrique},
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     volume = {7},
     year = {2023},
     doi = {10.46298/epiga.2023.volume7.7700},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/epiga.2023.volume7.7700/}
}
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Pirozhkov, Dmitrii. Stably semiorthogonally indecomposable varieties. Épijournal de Géométrie Algébrique, Tome 7 (2023). doi: 10.46298/epiga.2023.volume7.7700

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