Dolbeault Geometric Langlands Equivalence

• Dolbeault geometric Langlands equivalence is the classical-limit formulation that establishes a correspondence between the Higgs bundle moduli of a group and its Langlands dual via Hodge degeneration from D-modules to symbols.
• It employs nonabelian Hodge theory, Fourier–Mukai duality, and the framework of limit categories to rigorously address issues like singularities and non-quasi-compact moduli in Hitchin systems.
• The approach refines the correspondence through Whittaker normalization and semiorthogonal decompositions, extending its validity beyond the elliptic locus for groups such as GL_r, SL_r, and PGL_r.

Dolbeault Geometric Langlands Equivalence is the Dolbeault, or classical-limit, form of geometric Langlands. In its basic form, it identifies the Higgs-bundle sides for a complex reductive group $G$ and its Langlands dual $\check G$ by an equivalence

$$QC(T^*\operatorname{Bun}_G(X)) \simeq QC(T^*\operatorname{Bun}_{\check G}(X)),$$

viewed as the associated graded of de Rham geometric Langlands under the Hodge degeneration from $D$-modules to symbols. In later work, this naive cotangent-stack formulation is refined by replacing the automorphic side with a limit category on the full Higgs stack and the spectral side with coherent or ind-coherent sheaves on semistable Higgs bundles, in order to treat singularities, non-compactness, and non-quasi-compactness beyond the elliptic locus (Ben-Zvi et al., 2016, Pădurariu et al., 27 Aug 2025).

1. Classical-limit origin and nonabelian Hodge context

The standard background is Simpson’s triad of moduli problems for a smooth projective complex curve $X$ and a complex reductive group $\check G$: $$\operatorname{Conn}_{\check G}(X),\qquad \operatorname{Higgs}_{\check G}(X),\qquad \operatorname{Loc}_{\check G}(X).$$ The Riemann–Hilbert correspondence analytically identifies de Rham and Betti moduli, the de Rham space carries a nonabelian Hodge filtration whose special fiber is the Dolbeault space, and the nonabelian Hodge theorem gives a diffeomorphism between Dolbeault and de Rham moduli spaces after passing to semistable moduli spaces. This comparison package is the basic mechanism by which the Dolbeault form is understood as the classical limit of the de Rham form (Ben-Zvi et al., 2016).

On the automorphic side, $D(\operatorname{Bun}_G(X))$ carries a Hodge filtration obtained by degenerating differential operators to symbols, with special fiber

$QC(T^*\operatorname{Bun}_G(X)).$

Accordingly, the Dolbeault conjecture is presented as the associated graded analogue of the refined de Rham conjecture

$$D(\operatorname{Bun}_G(X)) \simeq QC^!_{\mathcal N}(\operatorname{Conn}_{\check G}(X)).$$

Donagi–Pantev, following an idea of Donagi and in a program pursued with Simpson, are explicitly singled out as using nonabelian Hodge theory on $\check G$0 to relate Higgs sheaves and $\check G$1-modules and thereby connect the de Rham and Dolbeault forms directly (Ben-Zvi et al., 2016).

2. Higgs stacks, Hitchin fibrations, and spectral curves

The geometric setting is the Hitchin system. For a reductive group $\check G$2, the derived moduli stack of Higgs bundles is

$\check G$3

and it carries the Hitchin map

$\check G$4

In the $\check G$5 case, a Higgs bundle is a pair $\check G$6 with $\check G$7, and the Hitchin base is

$\check G$8

For $\check G$9, the classical spectral curve is

$$QC(T^*\operatorname{Bun}_G(X)) \simeq QC(T^*\operatorname{Bun}_{\check G}(X)),$$0

with arithmetic genus

$$QC(T^*\operatorname{Bun}_G(X)) \simeq QC(T^*\operatorname{Bun}_{\check G}(X)),$$1

By the Beauville–Narasimhan–Ramanan correspondence, the Hitchin fiber over $$QC(T^*\operatorname{Bun}_G(X)) \simeq QC(T^*\operatorname{Bun}_{\check G}(X)),$$2 identifies with the moduli of torsion-free sheaves on $$QC(T^*\operatorname{Bun}_G(X)) \simeq QC(T^*\operatorname{Bun}_{\check G}(X)),$$3 with fundamental cycle $$QC(T^*\operatorname{Bun}_G(X)) \simeq QC(T^*\operatorname{Bun}_{\check G}(X)),$$4 (Toda, 27 Jun 2026).

The Hitchin map also governs the automorphic nilpotent condition. In the formulation emphasized in the Betti–de Rham–Dolbeault comparison,

$$QC(T^*\operatorname{Bun}_G(X)) \simeq QC(T^*\operatorname{Bun}_{\check G}(X)),$$5

and the global nilpotent cone is

$$QC(T^*\operatorname{Bun}_G(X)) \simeq QC(T^*\operatorname{Bun}_{\check G}(X)),$$6

It parameterizes $$QC(T^*\operatorname{Bun}_G(X)) \simeq QC(T^*\operatorname{Bun}_{\check G}(X)),$$7-bundles with nilpotent Higgs fields and is a conic Lagrangian substack. This structure later becomes the natural support condition in refined de Rham, Betti, and Dolbeault formulations (Ben-Zvi et al., 2016).

3. Fourier–Mukai, Hitchin fibers, and mirror-symmetry interpretation

The most concrete operational picture of Dolbeault geometric Langlands is fiberwise Fourier–Mukai duality along the Hitchin fibration. Donagi–Pantev are summarized as proving the Dolbeault conjecture over a dense open locus by reducing it to a Fourier–Mukai transform for abelian varieties applied to the fibers of Hitchin’s integrable system. On that locus, generic skyscrapers on

$$QC(T^*\operatorname{Bun}_G(X)) \simeq QC(T^*\operatorname{Bun}_{\check G}(X)),$$8

correspond to line bundles on smooth Hitchin fibers on the $$QC(T^*\operatorname{Bun}_G(X)) \simeq QC(T^*\operatorname{Bun}_{\check G}(X)),$$9-side (Ben-Zvi et al., 2016).

Kapustin–Witten provide the physical formulation behind this geometry. In their picture, the automorphic category is the category of $D$0-branes in the topological $D$1-model with target the Hitchin moduli space $D$2. A smooth Hitchin fiber is a Lagrangian torus, any rank-one local system on such a fiber defines an $D$3-brane, and T-duality along the Hitchin fibration sends it to a skyscraper $D$4-brane on the moduli of $D$5-local systems. These objects are Hecke or ’t Hooft eigenbranes. The same paper uses this Dolbeault picture to motivate the nilpotent singular-support condition: line bundles on Hitchin fibers have the global nilpotent cone as the support of their conical limit (Ben-Zvi et al., 2016).

The abelian toy model is the elliptic $D$6 case. For $D$7,

$D$8

the Hitchin system is projection to the second factor, and the self-duality of the Jacobian yields a fiberwise Fourier–Mukai auto-equivalence of $D$9 that exchanges a skyscraper on a fiber with a degree-zero line bundle on the same fiber. This model encapsulates the generic Hitchin-fiber mechanism in its simplest form (Ben-Zvi et al., 2016).

4. Refined formulation via limit categories

The naive equivalence between ordinary coherent categories on full Higgs stacks breaks down beyond the quasi-compact regime. For $X$0, $X$1, the full stacks

$X$2

are not compactly generated. Pădurariu–Toda therefore introduce limit categories for cotangent stacks of smooth stacks as an effective version of classical limits of categories of $X$3-modules (Pădurariu et al., 27 Aug 2025).

For a quasi-smooth derived stack $X$4 with self-dual cotangent complex, the category $X$5 is defined by weight-window conditions along all maps $X$6. For non-quasi-compact $X$7,

$X$8

with compact objects

$X$9

Applied to Higgs stacks, this produces the automorphic category $\check G$0 and its nilpotent refinement $\check G$1 (Pădurariu et al., 27 Aug 2025).

The refined Dolbeault geometric Langlands conjecture is then

$\check G$2

with compact form

$\check G$3

and nilpotent refinement

$\check G$4

The asymmetry is deliberate: semistable Higgs bundles appear on the spectral side, while the automorphic side uses the full Higgs stack only through the limit category. Pădurariu–Toda also prove that the automorphic limit category admits a semiorthogonal decomposition into quasi-BPS categories and construct Hecke operators on limit categories, expected to match Wilson operators under the conjectural equivalence (Pădurariu et al., 27 Aug 2025).

5. Proven cases and extensions beyond the elliptic locus

Before the recent limit-category developments, the best-understood region was the dense open locus of smooth or integral spectral curves, where fiberwise Fourier–Mukai methods on compactified Jacobians apply. The decisive shift occurs when reducible reduced curves are included, because the full Higgs stack then becomes genuinely non-quasi-compact (Ben-Zvi et al., 2016).

For $\check G$5, Toda proves the Dolbeault geometric Langlands correspondence over the open locus $\check G$6 of the Hitchin base where spectral curves are reduced: $\check G$7 This is described as the first non-trivial case in which the relevant moduli stacks are not quasi-compact and the use of limit categories is essential. Reduced spectral curves may still be reducible and singular. The proof uses the Arinkin Cohen–Macaulay extension of the Poincaré sheaf, a Fourier–Mukai transform, Wilson/Hecke compatibility, the Hitchin section, and the Whittaker normalization

$\check G$8

(Toda, 10 Feb 2026).

Toda then extends the theory in type $\check G$9 beyond the elliptic locus. For

$$\operatorname{Conn}_{\check G}(X),\qquad \operatorname{Higgs}_{\check G}(X),\qquad \operatorname{Loc}_{\check G}(X).$$0

the paper proves a Dolbeault geometric Langlands equivalence over an open locus strictly containing the elliptic locus, namely one containing the points where the spectral curve has at worst type $$\operatorname{Conn}_{\check G}(X),\qquad \operatorname{Higgs}_{\check G}(X),\qquad \operatorname{Loc}_{\check G}(X).$$1 singularities and allowing arbitrary numbers of irreducible components. For $$\operatorname{Conn}_{\check G}(X),\qquad \operatorname{Higgs}_{\check G}(X),\qquad \operatorname{Loc}_{\check G}(X).$$2, this includes generic reducible reduced curves such as unions of smooth components meeting transversely. In rank $$\operatorname{Conn}_{\check G}(X),\qquad \operatorname{Higgs}_{\check G}(X),\qquad \operatorname{Loc}_{\check G}(X).$$3,

$$\operatorname{Conn}_{\check G}(X),\qquad \operatorname{Higgs}_{\check G}(X),\qquad \operatorname{Loc}_{\check G}(X).$$4

so the result recovers the $$\operatorname{Conn}_{\check G}(X),\qquad \operatorname{Higgs}_{\check G}(X),\qquad \operatorname{Loc}_{\check G}(X).$$5 theorem and yields the Dolbeault geometric Langlands conjecture for $$\operatorname{Conn}_{\check G}(X),\qquad \operatorname{Higgs}_{\check G}(X),\qquad \operatorname{Loc}_{\check G}(X).$$6 over the reduced spectral-curve locus. The technical heart is the proof of Whittaker normalization over the type $$\operatorname{Conn}_{\check G}(X),\qquad \operatorname{Higgs}_{\check G}(X),\qquad \operatorname{Loc}_{\check G}(X).$$7-locus, together with the limit-category formalism required to control infinitely many Harder–Narasimhan strata (Toda, 27 Jun 2026).

6. Variants, comparisons, and open directions

A relative version of Dolbeault geometric Langlands has also been formulated for spherical varieties $$\operatorname{Conn}_{\check G}(X),\qquad \operatorname{Higgs}_{\check G}(X),\qquad \operatorname{Loc}_{\check G}(X).$$8 with abelian regular centralizers and no type $$\operatorname{Conn}_{\check G}(X),\qquad \operatorname{Higgs}_{\check G}(X),\qquad \operatorname{Loc}_{\check G}(X).$$9 roots. In that setting, the ambient ordinary Dolbeault equivalence for Hitchin systems is used to conjecture a Fourier–Mukai identification between a Dolbeault period sheaf and an explicit dual object built from the dual spherical group $D(\operatorname{Bun}_G(X))$0 and a dual symplectic representation $D(\operatorname{Bun}_G(X))$1. In polarized cases the dual object is a Dirac–Higgs bundle; in general it is an $D(\operatorname{Bun}_G(X))$2-sheaf defined from a cleaved cover determined by a symplectic Pfaffian divisor. This is not a new full categorical equivalence, but rather a precise Fourier–Mukai statement for distinguished objects, verified in cases such as the diagonal, Friedberg–Jacquet, Jacquet–Ichino, Rankin–Selberg, and Gross–Prasad examples (Hameister et al., 2024).

The Dolbeault form also sits within a larger de Rham–Betti–Dolbeault triangle. The Betti program explicitly treats Dolbeault eigensheaves as line bundles on Hitchin fibers, views the Betti automorphic category as an algebraic model for the $D(\operatorname{Bun}_G(X))$3-branes of Kapustin–Witten, and argues that cuspidal Hecke eigensheaves in the de Rham and Betti senses are expected to coincide. This suggests that Dolbeault geometry governs the semiclassical and mirror-symmetric regime, while de Rham and Betti versions package the same fundamental eigenobjects into different ambient categories (Ben-Zvi et al., 2016).

Several limitations remain explicit in the current literature. The cotangent-stack formulation

$D(\operatorname{Bun}_G(X))$4

still requires modification to account for singularities and non-compactness. The refined limit-category theory does not yet prove the conjecture on all of $D(\operatorname{Bun}_G(X))$5, nor for arbitrary reductive groups. The principal unresolved cases are non-reduced spectral curves, singularities beyond type $D(\operatorname{Bun}_G(X))$6, and settings where conductor subschemes are no longer curvilinear and the explicit resolution-and-weight-estimate mechanism for Arinkin-type kernels is unavailable. In this sense, the recent proofs establish that Dolbeault geometric Langlands extends beyond the elliptic locus, but only after replacing naive coherent categories by limit categories and proving Whittaker normalization in a form sensitive to non-quasi-compact Higgs geometry (Toda, 27 Jun 2026).