Christ’s conjecture on the equivalence between the Higgs category and the cosingularity category

Establish that, for the relative Ginzburg dg algebra A associated to the construction in this paper, the restriction Φ: H(A) → cosg(A) of the canonical quotient functor from the relative cluster category C(A) to the cosingularity category cosg(A) is an equivalence of k-linear categories, and show that for all objects X and Y in the Higgs category H(A) and all integers n ≥ 0, the induced maps Hom_H(X, Σ^{-n}Y) → Hom_cosg(A)(X, Σ^{-n}Y) are bijections.

Background

The paper constructs, for a simply laced Dynkin quiver Q, a relative 3-Calabi–Yau completion whose dg algebra A comes equipped with a relative cluster category C(A) (following Wu) and an associated Higgs category H(A). The cosingularity category cosg(A) is defined as per(A) modulo the localizing subcategory generated by all simple dg A-modules.

In Christ’s approach to categorifying the Goncharov–Shen moduli spaces for decorated surfaces, an expected key ingredient is that the Higgs category should coincide with the cosingularity category for the relevant Ginzburg algebras. The conjecture formulates this equivalence precisely, together with a prescribed compatibility on morphism spaces in non-positive degrees via bijections Hom_H(X, Σ^{-n}Y) ≅ Hom_cosg(A)(X, Σ^{-n}Y).