Stably semiorthogonally indecomposable varieties

TL;DR

This paper introduces the concept of noncommutatively stably semiorthogonally indecomposable (NSSI) varieties, showing their properties and implications for derived categories, including the absence of phantom subcategories in certain cases.

Contribution

It defines NSSI varieties, proves their stability under certain conditions, and applies this to show the nonexistence of phantom subcategories in specific varieties.

Findings

Varieties with finite Albanese morphism are NSSI.

Total spaces of fibrations over NSSI bases with NSSI fibers are NSSI.

No phantom subcategories in varieties like $C imes \\mathbb{P}^1$ for smooth proper curves $C$ of positive genus.

Abstract

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.