(Quasi-)affineness of perverse character varieties

Technical Summary: (Quasi-)Affineness of Perverse Character Varieties

Problem Statement
The paper addresses the geometric nature of perverse character varieties, which serve as moduli spaces for semisimple perverse sheaves on a stratified space $X$. While classical character varieties (parametrizing semisimple representations of a fundamental group) are known to be affine schemes, the affineness of their generalizations to perverse sheaves—specifically the "good moduli spaces" associated with perverse character stacks—had not been established in a purely stack-theoretic manner. The central problem is to determine whether these moduli spaces are quasi-affine, meaning they can be embedded as open subschemes of affine schemes.

Methodology
The authors employ a purely stack-theoretic approach, avoiding Geometric Invariant Theory (GIT) constructions. Their methodology relies on the following pillars:

1. Good Moduli Spaces: They utilize the theory of good moduli spaces for algebraic stacks (Alper, Halpern-Leistner, Heinloth). A good moduli space $q: \mathcal{X} \to X$ is an algebraic space that behaves like a categorical quotient, where points of $X$ correspond to isomorphism classes of polystable objects in $\mathcal{X}$.
2. Quasi-affineness Criterion: They establish that a good moduli space $X$ is quasi-affine if and only if its structure sheaf $\mathcal{O}_X$ separates polystable points (Corollary 2.1.6). This reduces the geometric problem to an algebraic one: constructing enough global sections of the structure sheaf to distinguish between non-isomorphic semisimple objects.
3. Hochschild Homology and Global Sections: The core technical innovation is the use of Hochschild homology ($HH$) of the stable $\infty$-category of constructible sheaves, denoted $\text{Cons}_P(X; \text{Mod}_k)$. Following the framework of Toën-Vaquié and Antieau-Gepner, the authors utilize the canonical morphism:
   $$HOC: HH(\mathcal{C}) \to R\Gamma(\mathcal{M}_{\mathcal{C}}, \mathcal{O}_{\mathcal{M}_{\mathcal{C}}})$$
   where $\mathcal{M}_{\mathcal{C}}$ is the moduli stack of objects in $\mathcal{C}$. This map allows the construction of global sections of the structure sheaf of the moduli stack directly from the Hochschild homology of the underlying category.
4. Recollement and Stratification: To handle the stratified nature of the space $X$, the authors use the recollement formalism for constructible sheaves. This allows them to decompose the Hochschild homology of the global category into a direct sum of Hochschild homologies of local systems on the strata (Lemma 4.3.1).
5. Trace Maps: The global sections constructed via $HOC$ are explicitly described as traces of monodromy. For a loop $\gamma$ in a stratum, the associated section evaluates a semisimple perverse sheaf $F$ to the trace of the monodromy of the restriction of $F$ to that stratum (Corollary 4.3.2).

Key Contributions and Results

• Main Theorem (Corollary 5.2.6): The authors prove that perverse character varieties are quasi-affine. Specifically, for a complex Whitney stratified space $X$ and a perversity function $p$, the canonical morphism
  $${}^p\text{Perv}_P(X) \to \text{Spec}(\Gamma({}^p\text{Perv}_P(X), \mathcal{O}_{{}^p\text{Perv}_P(X)}))$$
  is a quasi-compact open immersion. Here, ${}^p\text{Perv}_P(X)$ denotes the good moduli space (the perverse character variety).
• Separation of Points (Theorem 5.2.5): The paper demonstrates that the global sections derived from Hochschild homology are sufficient to separate points in the moduli stack. Specifically, if two semisimple perverse sheaves are non-isomorphic, there exists a global section (constructed from traces of monodromy on strata) that takes different values on them.
• Derived Setting: The results are established within the framework of derived algebraic geometry. The authors note that by Lurie's results, the derived and underived statements are equivalent, but working in the derived setting provides a natural framework for constructing the necessary sections.
• Generalization to Constructible Sheaves: The techniques are extended to the case of the "trivial perversity" (standard t-structure), proving that the good moduli space of constructible sheaves with flat stalks is also a disjoint union of quasi-affine schemes (Corollary 5.2.7).

Significance and Claims
The paper claims its primary contribution is providing an effective, intrinsic way to construct global sections of the structure sheaf of perverse character varieties without relying on GIT.

• "Beyond GIT": The authors emphasize that their techniques fall within the "Beyond GIT" moduli theory (referencing Alper-Halpern-Leistner), offering a more intrinsic approach to moduli problems that does not require a reductive group action or a specific linearization.
• Explicit Construction: Unlike abstract existence proofs, the paper provides a concrete description of the separating sections: they are traces of monodromy associated with loops in the strata of the underlying space.
• Scope: The results apply to any perversity for which the corresponding perverse character variety exists, including the middle perversity (central to the theory of D-modules and Riemann-Hilbert correspondence) and the trivial perversity.

The paper does not propose new experimental applications or future directions beyond the immediate mathematical context of establishing the quasi-affineness of these moduli spaces. It positions itself as a foundational result in the geometry of perverse sheaves, bridging the gap between the stack-theoretic description of moduli and their affine properties.