Dolbeault Geometric Langlands Conjecture

• The Dolbeault Geometric Langlands Conjecture is a formulation of geometric Langlands using the moduli stack of G-Higgs bundles as the classical limit in nonabelian Hodge theory.
• It employs the Hitchin integrable system to establish a fiberwise categorical equivalence between quasi-coherent sheaves on T*Bun_G and its Langlands dual counterpart.
• Recent advances validate the conjecture on dense open loci through limit-category frameworks that address non-quasi-compactness and impose nilpotent singular support.

The Dolbeault geometric Langlands conjecture is the Higgs-bundle, or “associated graded” / “classical limit,” form of geometric Langlands for a smooth projective complex curve $X$ and a complex reductive group $G$. In the nonabelian Hodge package, it places the moduli stack $\mathrm{Higgs}_G(X)$ of $G$-Higgs bundles alongside the de Rham moduli of flat connections and the Betti moduli of local systems, and asks for a Langlands-dual categorical equivalence governed by the Hitchin integrable system. In the survey formulation, this appears as an equivalence $$QC(T^*\mathrm{Bun}_G(X))\simeq QC(T^*\mathrm{Bun}_{G^\vee}(X))$$, while later work replaces this naive statement by a limit-category formalism adapted to singularities, nilpotent support, and the non-quasi-compactness of the full Higgs stack (Ben-Zvi et al., 2016, Pădurariu et al., 27 Aug 2025).

1. Conceptual position within nonabelian Hodge theory

For a smooth projective complex curve $X$, nonabelian Hodge theory studies three moduli problems for a complex reductive group: $\mathrm{Conn}_{G}(X)$, the moduli stack of flat $G$-connections; $\mathrm{Higgs}_{G}(X)$, the moduli stack of $G$-Higgs bundles; and $G$0, the moduli stack of $G$1-local systems. The de Rham and Betti moduli are related by the Riemann–Hilbert correspondence, which gives an analytic identification from the de Rham space to Betti space. The de Rham space also carries a nonabelian Hodge filtration, expressed via the Rees construction as a $G$2-equivariant family over $G$3 whose special fiber is the Dolbeault space. After passing from stacks to moduli spaces of semistable objects, the nonabelian Hodge theorem trivializes this filtration. In this sense, the Dolbeault theory is presented as the associated graded, or semiclassical, limit of the de Rham theory (Ben-Zvi et al., 2016).

A concise comparison is as follows.

Form	Moduli problem	Role in the package
Dolbeault	$G$4	graded / classical / Higgs realization
de Rham	$G$5	differential-operator / connection realization
Betti	$G$6	topological / local-system realization

This comparison is structural rather than rhetorical. The Dolbeault conjecture is explicitly described as the “classical limit” analogue of geometric Langlands, and Donagi–Pantev are credited with explaining how nonabelian Hodge theory on $G$7 should directly relate the de Rham conjecture to the Dolbeault conjecture (Ben-Zvi et al., 2016).

A persistent source of confusion is that the proof of the de Rham geometric Langlands conjecture in the Arinkin–Gaitsgory form does not, by itself, prove the Dolbeault conjecture. The final paper in the de Rham proof series is explicit that its argument is specific to the de Rham setting, even though it fits the same broader spectral philosophy (Gaitsgory et al., 2024). This suggests that the Dolbeault problem requires its own categorical and geometric apparatus.

2. The classical Dolbeault statement and Hitchin geometry

In its original surveyed form, the Dolbeault geometric Langlands conjecture is stated as an equivalence of dg categories

$G$8

Here the Dolbeault automorphic category is the category of quasi-coherent Higgs sheaves on the cotangent stack to $G$9, obtained as the special fiber of the Hodge-filtration degeneration of $\mathrm{Higgs}_G(X)$0-modules to symbols. The conjecture is presented as the Higgs-bundle version of geometric Langlands, and as the associated graded limit of the de Rham correspondence (Ben-Zvi et al., 2016).

The geometric backbone of this formulation is the Hitchin integrable system

$\mathrm{Higgs}_G(X)$1

induced by $\mathrm{Higgs}_G(X)$2. Its zero fiber

$\mathrm{Higgs}_G(X)$3

is the global nilpotent cone. In the surveyed picture, Hitchin fibers are the basic Dolbeault spectral objects, and the conjectural duality is interpreted fiberwise along the Hitchin fibration. The paper emphasizes that the conjecture is compatible with suitable Dolbeault Hecke functors, but that it must be modified to account for singularities and non-compactness on both sides (Ben-Zvi et al., 2016).

The clearest toy model is the elliptic curve. There the Dolbeault space is

$\mathrm{Higgs}_G(X)$4

with Hitchin map given by projection to the second factor. In this model, the Fourier–Mukai transform exchanges a skyscraper sheaf on a fiber with a degree-zero line bundle on the same fiber. This is the prototype of the Dolbeault Langlands transform: fiberwise abelian duality inside the Hitchin system (Ben-Zvi et al., 2016).

The same survey records the first partial result: Donagi–Pantev proved the 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 one side correspond to line bundles on smooth Hitchin fibers on the dual side (Ben-Zvi et al., 2016).

3. Why the naive formulation is insufficient

The modern consensus in the cited literature is that the naive form

$\mathrm{Higgs}_G(X)$5

is not the correct global Dolbeault statement. The reason given is geometric rather than philosophical: $\mathrm{Higgs}_G(X)$6 is typically non-quasi-compact, its coherent-type categories fail to be compactly generated, and outside the elliptic locus the full Higgs stack contains infinitely many Harder–Narasimhan strata (Pădurariu et al., 27 Aug 2025).

To correct this, Toda–Pădurariu introduce limit categories for cotangent stacks of smooth stacks as an effective version of classical limits of categories of $\mathrm{Higgs}_G(X)$7-modules. For a cotangent stack $\mathrm{Higgs}_G(X)$8, the relevant category is denoted $\mathrm{Higgs}_G(X)$9, and for non-quasi-compact stacks its ind-completion is defined by Zariski gluing over quasi-compact open substacks. These limit categories are designed to be much smaller than $G$0, while retaining the formal behavior expected of the classical limit of coherent $G$1-modules. In particular, the paper presents

$G$2

as the guiding classical-limit passage (Pădurariu et al., 27 Aug 2025).

In this framework the precise Dolbeault geometric Langlands conjecture becomes

$G$3

and on compact objects

$G$4

Here $G$5, and $G$6 determines the twisting on the automorphic side via a $G$7-line bundle $G$8 on $G$9. The left-hand side is the derived category of coherent sheaves on the semistable Higgs moduli stack for the Langlands dual group; the right-hand side is the compactly generated limit category attached to the full Higgs stack of the original group (Pădurariu et al., 27 Aug 2025).

The functorial structure is part of the point. The same paper proves smooth pull-back and projective push-forward for limit categories, establishes compatibility with composition and base change, and, crucially, constructs a left adjoint $$QC(T^*\mathrm{Bun}_G(X))\simeq QC(T^*\mathrm{Bun}_{G^\vee}(X))$$0 for suitable open immersions. This is the mechanism that makes the non-quasi-compact Higgs stack usable in a Dolbeault formulation beyond the elliptic locus (Pădurariu et al., 27 Aug 2025).

4. Refined categorical structures: nilpotent support, quasi-BPS categories, and Eisenstein theory

The refined Dolbeault picture is not only about replacing $$QC(T^*\mathrm{Bun}_G(X))\simeq QC(T^*\mathrm{Bun}_{G^\vee}(X))$$1 by a smaller category; it also imports the support-theoretic and parabolic structures familiar from modern geometric Langlands. In the broader singular-support formalism for quasi-smooth derived stacks, $$QC(T^*\mathrm{Bun}_G(X))\simeq QC(T^*\mathrm{Bun}_{G^\vee}(X))$$2, so singular support measures precisely the failure of an ind-coherent object to be quasi-coherent. In geometric Langlands, nilpotent singular support was introduced as the correct intermediate size between $$QC(T^*\mathrm{Bun}_G(X))\simeq QC(T^*\mathrm{Bun}_{G^\vee}(X))$$3 and all of $$QC(T^*\mathrm{Bun}_G(X))\simeq QC(T^*\mathrm{Bun}_{G^\vee}(X))$$4 on singular spectral stacks (Arinkin et al., 2012). Modern Dolbeault formulations follow the same pattern.

In the limit-category approach, the compact part of the Dolbeault category admits a semiorthogonal decomposition into quasi-BPS categories. For $$QC(T^*\mathrm{Bun}_G(X))\simeq QC(T^*\mathrm{Bun}_{G^\vee}(X))$$5, the compact limit category $$QC(T^*\mathrm{Bun}_G(X))\simeq QC(T^*\mathrm{Bun}_{G^\vee}(X))$$6 is compactly generating, and its compact objects have a semiorthogonal decomposition indexed by standard parabolics $$QC(T^*\mathrm{Bun}_G(X))\simeq QC(T^*\mathrm{Bun}_{G^\vee}(X))$$7 with Levi quotient $$QC(T^*\mathrm{Bun}_G(X))\simeq QC(T^*\mathrm{Bun}_{G^\vee}(X))$$8: $$QC(T^*\mathrm{Bun}_G(X))\simeq QC(T^*\mathrm{Bun}_{G^\vee}(X))$$9 The quasi-BPS summands are presented as categorical versions of BPS invariants on a non-compact Calabi–Yau $X$0-fold, and the conjectural Dolbeault equivalence is required to match these semiorthogonal decompositions. In type $X$1, this is interpreted as a categorical form of the Hausel–Thaddeus topological mirror symmetry picture (Pădurariu et al., 27 Aug 2025).

Hecke symmetries are built into this formulation. The limit-category paper constructs Hecke operators on $X$2; for $X$3 they come from a monoidal category

$X$4

built from quasi-BPS categories of zero-dimensional sheaves on the local surface $X$5, and for general reductive $X$6 the paper constructs Hecke functors

$X$7

for minuscule coweights $X$8. These Hecke operators are expected to match Wilson operators on the dual side under Dolbeault geometric Langlands (Pădurariu et al., 27 Aug 2025).

A different but complementary refinement appears in work on geometric Eisenstein series in nonabelian Hodge theory. There, Dolbeault, Hodge, and twistor Eisenstein functors are constructed on coherent nilpotent sheaves, and the resulting categories admit cuspidal–Eisenstein decompositions. In particular, $X$9, $\mathrm{Conn}_{G}(X)$0, and the twistor BBB categories are generated by the essential images of Eisenstein functors from proper Levi subgroups, with the quotient identified with perfect complexes on the irreducible locus. This is presented as the Dolbeault analogue of the Arinkin–Gaitsgory spectral decomposition and as evidence that the correct Dolbeault spectral categories should be built from nilpotent sheaves and parabolic induction (Hanson, 30 Dec 2025).

5. Proven cases and the current state of the conjecture

The oldest positive result recorded in the cited literature is the Donagi–Pantev theorem over a dense open locus, where the conjecture reduces to Fourier–Mukai duality for abelian varieties on smooth Hitchin fibers (Ben-Zvi et al., 2016). This is a genuine theorem, but only over the locus where the Hitchin system is sufficiently regular.

A major recent advance is the proof for type $\mathrm{Conn}_{G}(X)$1 groups beyond the elliptic locus. For $\mathrm{Conn}_{G}(X)$2, $\mathrm{Conn}_{G}(X)$3, or $\mathrm{Conn}_{G}(X)$4 with $\mathrm{Conn}_{G}(X)$5, there is an open subset

$\mathrm{Conn}_{G}(X)$6

strictly containing the elliptic locus such that the Dolbeault geometric Langlands conjecture, in the sense of Toda–Pădurariu, holds over $\mathrm{Conn}_{G}(X)$7. Here $\mathrm{Conn}_{G}(X)$8 is the locus where spectral curves have at worst type $\mathrm{Conn}_{G}(X)$9 singularities. For $G$0, the theorem is stated over $G$1; for the dual pair $G$2, it is proved over $G$3, where $G$4 is the traceless Hitchin base (Toda, 27 Jun 2026).

This proof uses the nilpotent-support version of the Dolbeault equivalence. The role of limit categories is essential outside the elliptic locus because the full Higgs stack is no longer quasi-compact, stable and semistable loci do not coincide with the full stack, and there are infinitely many Harder–Narasimhan strata (Toda, 27 Jun 2026).

The technical heart of the proof is the Whittaker normalization conjecture. Over the type $G$5 locus, the authors prove that the Fourier–Mukai functor defined by the Arinkin sheaf sends the structure sheaf on the semistable dual Hitchin stack to the $G$6-extension of the structure sheaf of the Hitchin section. The proof proceeds through explicit resolutions of the Arinkin sheaf on type $G$7 spectral curves, $G$8-weight estimates, local cohomology vanishing along Harder–Narasimhan strata, perturbation of stability after étale base change, reduction to fine compactified Jacobians, and normalization along the Hitchin section. As a consequence, the paper also obtains the conjecture for $G$9 over the reduced spectral curve locus (Toda, 27 Jun 2026).

These results sharpen two important points. First, the conjecture is not restricted to integral or elliptic spectral curves. Second, the correct categorical statement beyond the elliptic locus is not ordinary coherent duality on full Higgs stacks, but a limit-category equivalence with nilpotent support conditions (Toda, 27 Jun 2026).

6. Relative, twistor, and analytic extensions

The Dolbeault perspective extends beyond the absolute case of $\mathrm{Higgs}_{G}(X)$0. For an affine homogeneous spherical variety $\mathrm{Higgs}_{G}(X)$1 with abelian regular centralizers and no type $\mathrm{Higgs}_{G}(X)$2 roots, a relative Dolbeault geometric Langlands conjecture is formulated for $\mathrm{Higgs}_{G}(X)$3. The relevant moduli stack is the Dolbeault moduli of $\mathrm{Higgs}_{G}(X)$4-Higgs bundles

$\mathrm{Higgs}_{G}(X)$5

equipped with a Hitchin-type map to an affine base $\mathrm{Higgs}_{G}(X)$6. The key $\mathrm{Higgs}_{G}(X)$7-side object is the Dolbeault period sheaf, while the $\mathrm{Higgs}_{G}(X)$8-side object is a sheaf built from the dual symplectic representation $\mathrm{Higgs}_{G}(X)$9, either as a Dirac–Higgs bundle in polarized cases or as a pushforward along a cleaved cover in general. The invariant-theoretic core is a matching-divisors statement identifying the non-separated divisor of the regular quotient with the Pfaffian divisor of $G$0. This framework is verified in several cases, including Friedberg–Jacquet, Rankin–Selberg, Jacquet–Ichino, and Gross–Prasad (Hameister et al., 2024).

A second extension places Dolbeault geometric Langlands inside the Hodge and twistor families of nonabelian Hodge theory. The Hodge stack interpolates between $G$1 and $G$2, and the twistor stack produces a $G$3-family with special fibers $G$4 and $G$5 and generic fiber $G$6. Within this framework, Dolbeault, Hodge, and twistor Eisenstein functors are constructed, BBB-brane categories are defined as rigid subcategories of twistor ind-coherent sheaves, and a twistor geometric Langlands conjecture

$G$7

is proposed as a mathematical form of the Kapustin–Witten BBB/BAA duality. The paper treats the Dolbeault conjecture as the $G$8 fiber, or classical limit, of this larger twistor package (Hanson, 30 Dec 2025).

There is also an analytic avatar. A function-theoretic version for complex curves replaces Hecke eigensheaves by joint eigenfunctions of commuting global differential operators—quantum Hitchin Hamiltonians and their complex conjugates—acting on a Hilbert space of half-densities on $G$9. Using the Beilinson–Drinfeld identification

$G$00

the paper conjectures a canonical self-adjoint extension of the symmetric real operator algebra and identifies its spectrum with opers satisfying a reality condition. This is proved for $G$01 and for the simplest non-abelian case $G$02 on $G$03 with four marked points. The paper is explicit that this is an analytic shadow of the Hitchin-side Langlands picture rather than a proof of the Dolbeault conjecture itself (Etingof et al., 2019).

Taken together, these developments fix the present mathematical profile of the Dolbeault geometric Langlands conjecture. It is the Higgs-bundle realization of geometric Langlands; it is organized by the Hitchin system and its fiberwise Fourier–Mukai dualities; it is no longer expected to be adequately expressed by naive $G$04-equivalences on full Higgs stacks; and its modern forms involve semistable Higgs moduli, limit categories, nilpotent support, Hecke/Wilson compatibility, and parabolic decompositions. Partial proofs now exist on substantial open loci for type $G$05, while relative, Hodge, twistor, and analytic versions indicate that the conjecture belongs to a larger nonabelian Hodge and mirror-symmetric framework rather than to a single isolated categorical equivalence (Pădurariu et al., 27 Aug 2025, Toda, 27 Jun 2026).