Orbifold Semiorthogonal Decompositions for Abelian Varieties
Bronson Lim, Franco Rota
备注：
v1: 21 pages, comments welcome! v2: 25 pages. Title changed, details added in Section 4, improved overall exposition and references updated. Accepted manuscript, to appear in J. noncommut. geom
发表信息：
J. Noncommut. Geom. (2024), published online first
Suppose $G$ is a finite group acting on an Abelian variety $A$ such that
the coarse moduli space $A/G$ is smooth. Using the recent classification result due to
Auffarth, Lucchini Arteche, and Quezada, we construct an orbifold semiorthogonal decomposition for
$\mathcal{D}[A/G]$ provided $G = T\rtimes H$ with $T$ a subgroup of translations and $H$ is a subgroup of group automorphisms.
In this short note, we observe that Theorem 3.1 in arXiv:1508.00682 for semiorthogonal indecomposability of the derived category of smooth DM stacks based on the canonical bundle can be extended to the case of projective varieties with Cohen-Macaulay singularities. As a consequence, all projective curves of positive arithmetic genus have weakly indecomposable bounded derived categories and indecomposable categories of perfect complexes. Here weak indecomposablility refers to the admissibility of components.
Given a smooth variety $X$ with an action of a finite group $G$, and a semiorthogonal decomposition of the derived category, $\mathcal{D}([X/G])$, of $G$-equivariant coherent sheaves on $X$ into subcategories equivalent to derived categories of smooth varieties, we construct a similar semiorthogonal decomposition for a smooth $G$-invariant divisor in $X$ (under certain technical assumptions). Combining this procedure with the semiorthogonal decompositions constructed in [PV15], we construct semiorthogonal decompositions of some equivariant derived categories of smooth projective varieties.
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