Stably semiorthogonally indecomposable varieties

Abstract: A triangulated category is said to be indecomposable if it admits no nontrivial semiorthogonal decompositions. For a derived category of coherent sheaves on a variety Y, we propose a stronger condition, which implies, among other things, that for any variety X, any semiorthogonal decomposition of the product X x Y is induced from a decomposition of X. For X = {pt} this implies the usual indecomposability. We show that varieties with finite Albanese morphism, e.g., curves of positive genus, are stably semiorthogonally indecomposable in this sense. From this, we deduce the non-existence of phantom subcategories in the product surfaces C x P^1, where C is a smooth projective curve of positive genus, and in some other examples as well.

algebraic geometry

Audience: researchers in the topic

Derived seminar

Series comments: https://ed-ac-uk.zoom.us/j/89993982042

Password: a simply-connected two-dimensional variety with trivial canonical bundle (omit the space)