The Dolbeault geometric Langlands correspondence for type A groups beyond the elliptic locus

Abstract: In this paper, we prove a Dolbeault geometric Langlands equivalence for $\GL_r$ and for the Langlands dual pair $\SL_r/\PGL_r$ over an open locus of the Hitchin base which strictly contains the elliptic locus. This open locus contains the points corresponding to spectral curves with at worst type $A$ singularities, without any restriction on the number of irreducible components. The Dolbeault geometric Langlands equivalence considered here is the one formulated in our previous work with Tudor Pădurariu, which links categorical Donaldson--Thomas theory with the geometric Langlands correspondence. It relates coherent sheaves on moduli stacks of semistable Higgs bundles to the limit category associated with the full moduli stack of Higgs bundles. The use of limit categories is essential beyond the elliptic locus, where the full Higgs moduli stack is no longer quasi-compact and contains infinitely many Harder--Narasimhan strata. The key step is to prove the Whittaker normalization conjecture over the locus of spectral curves with type $A$ singularities, following and extending the strategy developed in the author's proof of the $\GL_2$ case over the reduced spectral curve locus. As a consequence, we also obtain the Dolbeault geometric Langlands conjecture for $\SL_2/\PGL_2$ over the reduced spectral curve locus.

• The paper establishes a categorical equivalence of Higgs moduli limit categories for GL_r, SL_r, and PGL_r across spectral curves beyond the elliptic locus.
• It introduces explicit resolutions, weight calculations, and étale base change techniques to manage type A singularities in non-quasi-compact settings.
• The work advances the geometric Langlands program by linking categorical Donaldson–Thomas theory with derived moduli invariants in highly singular contexts.

The Dolbeault Geometric Langlands Correspondence for Type A Groups Beyond the Elliptic Locus

Introduction and Context

This paper addresses the Dolbeault incarnation of the geometric Langlands program for groups of type $A$, specifically $GL_r$, $SL_r$, and $PGL_r$, moving substantially beyond the domain where the objects under consideration (namely, spectral curves) are integral. Traditionally, derived equivalences of categories of sheaves on moduli stacks of Higgs bundles (the Dolbeault side) and their Langlands duals have been established only on the so-called "elliptic locus," corresponding to moduli supported over irreducible and reduced spectral curves. This work rigorously extends the equivalence to a much larger open subset of the Hitchin base, including all spectral curves presenting at worst type $A$ singularities, regardless of the number of components or global reducibility.

The geometric structures at play are highly singular and non-compact outside the elliptic locus: Higgs moduli stacks acquire infinitely many Harder–Narasimhan strata, are not quasi-compact, and lose basic finiteness properties. The main results are enabled by the framework of "limit categories" and techniques at the interface of categorical Donaldson–Thomas theory and derived algebraic geometry.

Main Results

The primary theorem establishes the Dolbeault geometric Langlands equivalence for $GL_r$, $SL_r$, and $PGL_r$ over an explicit open locus of the Hitchin base that strictly contains the elliptic locus. The open subset includes spectral curves with at worst type $A$ singularities (locally $y^2 = x^m$ at singular points), without any restriction on the number of irreducible components. Explicitly, a $GL_r$0 is constructed so that the conjectural categorical equivalence—relating limit categories of coherent sheaves on the moduli stack of semistable Higgs bundles for the Langlands dual pair—holds over $GL_r$1.

A significant corollary is the full proof of the Dolbeault geometric Langlands correspondence for $GL_r$2 over the locus of reduced spectral curves (for $GL_r$3, type-A singularities precisely coincide with reducedness).

The categorical meaning of the equivalence involves a nontrivial technical innovation: over the non-elliptic locus, one must work with appropriately constructed limit categories that rigidly capture the correct "boundary conditions" in the non-quasi-compact case. These categories provide a categorified version of classical limits of $GL_r$4-modules on $GL_r$5 and naturally interface with BPS sheaves arising from Donaldson–Thomas theory.

Methodology and Technical Contributions

The proof relies critically on the validation of the so-called "Whittaker normalization conjecture" for points of the Hitchin base underlying spectral curves with type $GL_r$6 singularities. This conjecture is technically nontrivial; it asserts an isomorphism between the action of the (Arinkin's) Cohen–Macaulay extension of the Poincaré line bundle via a Fourier–Mukai transform and a certain left adjoint functor ($GL_r$7) corresponding to the Hitchin section.

The proof proceeds in several stages:

1. Explicit Resolutions and Weight Calculations: The author constructs explicit left resolutions of Arinkin-type sheaves $GL_r$8 for semistable rank-one torsion-free sheaves $GL_r$9 on spectral curves with type $SL_r$0 singularities. These are resolved in terms of explicit vector bundles, and the key step is a detailed analysis of their $SL_r$1-weights under all possible Harder–Narasimhan degenerations—these weights encode the behavior with respect to functoriality required by the limit category formalism.
2. Verification of Categorical Boundary Conditions: Using the weight calculations, the author shows that the relevant Arinkin sheaves belong to the subcategory defined by the limit condition (as per the intrinsic magic window property). This is subtle and essential, particularly due to the infinite type of the stacks outside the elliptic locus.
3. Use of Deformations and Étale Base Changes: To address the noncompactness and refine stability conditions, the argument exploits étale base changes of the Hitchin base allowing local perturbations of the stability condition. This reduction to almost proper cases, in combination with derived equivalence results for fine compactified Jacobians of planar curves, ensures that the required equivalence can be checked scheme-theoretically in finite type neighborhoods.
4. Descent to Dual Pairs and Functorial Compatibility: By an explicit descent process, making use of the norm maps and properties of the Poincaré sheaf, the results are extended from $SL_r$2 to $SL_r$3 and $SL_r$4. Diagrammatic arguments confirm that all functors intertwine as needed to induce equivalences of the relevant limit categories.
5. Control over Singular Supports and Local-to-Global Arguments: Technical lemmas confirm that all relevant functors preserve nilpotent singular support, and conversely, that conservativity of certain pullbacks/pushforwards ensures full faithfulness of the equivalence on limit categories.

Implications and Theoretical Advances

The extension of the Dolbeault geometric Langlands equivalence beyond the elliptic locus is a major advance, substantially enlarging the domain over which the geometric Langlands philosophy can be rigorously formulated and proved. The techniques developed—especially the systematic use of limit categories and weight windows—provide a robust framework for categorical equivalences in highly singular and noncompact settings, with potential applicability far beyond the specific case of type $SL_r$5 groups.

Practically, the inclusion of spectral curves with arbitrarily complicated reducibility and milder singularities opens the correspondence to moduli points relevant in wall-crossing phenomena, intersection theory, and the study of categorical Donaldson–Thomas invariants. The approach also clarifies and resolves technical obstacles to pushing geometric representation theory into genuinely global, singular, and non-finite type regimes.

On the theoretical side, these results shed new light on the relationships between:

• Categorified Donaldson–Thomas theory,
• Categorical and homological aspects of the geometric Langlands program,
• The structure of (possibly infinite type) moduli stacks and their associated (co)homological invariants.

Potential Future Developments

The methods and results of this work suggest several directions for future research:

• Extension to More General Singularities:

While type $SL_r$6 singularities represent a broad generalization, further progress may be possible towards other classes of singularities (including non-planar ones), or perhaps to even larger open loci in the Hitchin base.

• Other Reductive Groups:

The techniques may be adaptable to higher type ($SL_r$7, $SL_r$8, $SL_r$9, etc.) groups, pending the appropriate spectral data and compact generation properties of the resulting limit categories.

• Interaction with Physical Theories:

The categorical and weight window techniques resonate with structures arising in physics, suggesting further investigation of their role in derived categories of branes, wall-crossing, and moduli of vacua.

• Refinements in Categorical Donaldson–Thomas and BPS Invariants:

The explicit realization of the limit category as a dual notion to Higgs bundle semistability could lead to deeper structural relationships with BPS algebras and wall-crossing functors.

Conclusion

This work proves the Dolbeault geometric Langlands equivalence for type $PGL_r$0 groups well beyond previously accessible loci, by complete development of the required categorical, homological, and combinatorial tools. It demonstrates the power and flexibility of limit categories in the face of the serious challenges posed by non-finite type algebraic stacks and their moduli and deepens the categorical interface between geometric representation theory and Donaldson–Thomas theory.