Admissible subcategories supported on curves

TL;DR

This paper investigates the structure of admissible subcategories supported on curves within smooth projective varieties, revealing conditions on their components and implications for the derived category's indecomposability.

Contribution

It proves that irreducible components of support are rational curves and establishes conditions linking these components to the canonical class, confirming a conjecture for surfaces.

Findings

Irreducible components of support are rational curves.

At least one component intersects the canonical class negatively.

Surfaces with nef canonical bundle have indecomposable derived categories.

Abstract

Let $X$ be a smooth projective variety. We study admissible subcategories of the bounded derived category of coherent sheaves on $X$ whose support is a proper subvariety $Z \subset X$. We show that any one-dimensional irreducible component of $Z$ is a rational curve. When $\operatorname{dim} Z = 1$, we prove that at least one irreducible component in $Z$ intersects the canonical class $K_X$ negatively. In particular, this implies that a surface with a nef and effective canonical bundle has indecomposable derived category, confirming the conjecture by Okawa. We also prove that a configuration of curves with non-negative self-intersections on a surface cannot support an admissible subcategory.