Silting reduction, relative AGK's construction and Higgs construction
Yilin Wu
TL;DR
The paper generalizes Iyama–Yang's Calabi--Yau triple to a $(d+1)$-Calabi--Yau quadruple and proves the associated Higgs category is a $d$-Calabi--Yau Frobenius extriangulated category with a canonical $d$-cluster-tilting subcategory. It develops the relative Amiot--Guo--Keller construction and the Higgs construction for such quadruples, establishing equivalences with relative cluster categories and describing stability hearts. It shows that silting reduction interacts compatibly with CY reduction, yielding reduced quadruples and equivalences between Higgs categories and their quotients up to necessary subcategories. The results encompass concrete instances from ice quivers with potentials and isolated singularities, and provide a unified framework for reductions and equivalences in higher-dimensional Calabi--Yau context, with dg-enhanced realizations.
Abstract
We introduce the notion of a Calabi--Yau quadruple as a generalization of Iyama--Yang's Calabi--Yau triple. For each $(d+1)$-Calabi--Yau quadruple, we show that the associated Higgs category is a $d$-Calabi--Yau Frobenius extriangulated category, which moreover admits a canonical $d$-cluster-tilting subcategory. Concrete examples arise from the construction of relative cluster categories and Higgs categories in the setting of ice quivers with potentials, as well as from the singularity category of an isolated singularity. As an application, we prove that both the relative Amiot--Guo--Keller's construction and the Higgs construction of a $(d+1)$-Calabi--Yau quadruple take silting reduction to Calabi--Yau reduction.