An $L^2$-cohomology theory for harmonic Higgs bundles on arbitrary Galois coverings of Zariski open sets

Dingoyan and Schumacher (2018) proved, under a Poincaré covering hypothesis — bounded geometry and positive injectivity radius of the pulled-back Poincaré-type metric — that the $L^2$-cohomology of harmonic Higgs bundles on Galois coverings of Zariski open subsets of compact Kähler manifolds carries a full Hodge-theoretic package: $\Gamma$-Fredholmness of the relevant Laplacians, an index formula in terms of logarithmic characteristic integrals, degeneration at $E_2$ of the $L^2$ Dolbeault-to-de-Rham spectral sequence after tensoring with the algebra of affiliated operators, and comparison with $L^2$ simplicial cohomology. The authors themselves acknowledge that the hypothesis is a strong constraint that is not easy to check, that the results fail verbatim for arbitrary Galois coverings, and they announce a forthcoming model treating arbitrary coverings — a model that never appeared in print. The problem is to construct an $L^2$ Hodge theory for harmonic Higgs bundles on arbitrary Galois coverings of Zariski open sets, recovering suitable analogues of all of the above results. The problem remains open: partial progress exists only in dimension one for variations of Hodge structure on coverings ramified at infinity (Jean 2022), while the related program of Eyssidieux (2022) for Hodge modules on coverings of compact manifolds remains conjectural.

Background

Let $\bar{Y}$ be a compact Kähler manifold and $D$ a normal crossings divisor, i.e. a divisor locally of the form $\{z_1\cdots z_k=0\}$ in holomorphic coordinates; the complement $Y=\bar{Y}\setminus D$ is a typical Zariski open (quasi-projective) manifold. Such a $Y$ carries a complete Kähler metric $\omega_{\bar{Y},D}$ of finite volume with cuspidal behavior near $D$ (locally built from $-\log\log|z_i|$), called a Poincaré-type metric. Given a Galois covering $p:X\to Y$ — an unramified covering whose deck group $\Gamma$ acts transitively on fibers — one pulls the metric back to $X$ and studies the $L^2$-cohomology: the cohomology of the complex of square-integrable differential forms whose differentials are again square-integrable. Because $\Gamma$ acts on everything, the spaces of $L^2$-harmonic forms are modules over the group von Neumann algebra $M(\Gamma)$ and carry a Murray–von Neumann dimension $\dim_\Gamma$ — finite under the Poincaré covering hypothesis — the basic numerical invariant. A $\Gamma$-equivariant operator $A$ is called $\Gamma$-Fredholm when, for some $\epsilon>0$, the spectral projections of $AA^*$ and $A^*A$ onto $[0,\epsilon]$ have Schwartz kernels integrable over a fundamental domain; this is the finiteness condition that makes the theory workable. Classical Cheeger–Gromov theory extends the $L^2$-index theorem for coverings of compact manifolds to coverings of complete finite-volume manifolds of bounded curvature, provided the covering itself has bounded geometry (curvature and all its covariant derivatives bounded) and positive injectivity radius (a uniform lower bound on the radius of geodesic balls on which the exponential map is injective).

Dingoyan and Schumacher (2018, an unpublished preprint) review this theory and apply it to harmonic Higgs bundles. A Higgs bundle is a holomorphic vector bundle $E$ with a holomorphic one-form-valued endomorphism $\theta$ (the Higgs field) satisfying $\theta\wedge\theta=0$; it is harmonic when it carries a Hermitian metric $h$ solving the equations of nonabelian Hodge theory, under which the associated connection $\nabla=\bar\partial_E+\partial_E+\theta+\theta^{*}$ is flat. By the nonabelian Hodge correspondence, harmonic Higgs bundles on $Y$ are the analytic avatar of complex reductive linear representations of $\pi_1(Y)$; tame nilpotent refers to the standard polynomial growth condition on $\theta$ near $D$ under which harmonic metrics are known to exist with controlled asymptotics. Under the hypothesis that $p$ is a Poincaré covering — meaning that the pull-back of $\omega_{\bar{Y},D}$ to $X$ has bounded geometry and positive injectivity radius, a condition automatic for coverings whose monodromy around $D$ produces cusps but failing for coverings ramified at infinity — the paper proves: (i) the Kodaira Laplacian (the $\bar\partial$-Laplace operator) and the Higgs Laplacian (its deformation by the Higgs field), acting on bundles of bounded curvature, are $\Gamma$-Fredholm, so the $L^2$-Betti numbers are finite; (ii) a Galois $\partial\bar\partial$-lemma, the $L^2$ covering analogue of the classical compact-Kähler fact that a form which is $\partial$- and $\bar\partial$-closed and exact in either sense is already $\partial\bar\partial$-exact; (iii) index formulas identifying the $L^2$-Euler characteristic with the logarithmic Euler characteristic of $(\bar{Y},D)$ and, for tame nilpotent harmonic Higgs bundles of rank $r$, with $r$ times an integral of Todd and Chern classes of the logarithmic tangent and sheaves of logarithmic forms; (iv) degeneration at $E_2$ of the $L^2$ Dolbeault-to-de-Rham spectral sequence (a spectral sequence computes a target cohomology through successive pages $E_r$; degeneration at $E_2$ means the pages stabilize and give the answer exactly from the second page on) $(H^{p,q}_{\bar\partial(2)}(X,E),\theta)\otimes_{M(\Gamma)}\mathcal{U}(\Gamma)\Rightarrow H^{p+q}_{\nabla(2)}(X,E)\otimes_{M(\Gamma)}\mathcal{U}(\Gamma)$, where $\mathcal{U}(\Gamma)$ is the algebra of affiliated (possibly unbounded) operators of $M(\Gamma)$, in which von-Neumann-algebraic torsion is inverted; and (v) a homotopy-invariance theorem identifying the $L^2$-Betti numbers of $(X,E)$ with those of the pull-back simplicial complex with coefficients in the locally constant sheaf of the flat bundle, under a bundle hypothesis called $AC^0$ (the $k=0$ reading of Definition 2.2.2 of the paper, which defines $AC^k$ only for $1\le k\in\mathbb{N}$: there exist a metric $h'$ with metric connection $\nabla_{h'}$ for which the bundle has bounded geometry, strictly positive constants $c_1,c_2$ with $c_1h\le h'\le c_2h$, the difference connection form $\nabla_h-\nabla_{h'}$ has bounded covariant derivatives up to order $k-1$ — a condition vacuous at $k=0$ — and the symmetries of $(E,h,\nabla_h)$ are required to be symmetries of $(E,h',\nabla_{h'})$, which in the covering setting means $\Gamma$-equivariance). Dingoyan's earlier Math. Ann. paper (2013) had obtained the companion mixed Hodge structure on such $L^2$-cohomology groups, in quotient categories after inverting $\mathcal{U}(\Gamma)$-torsion, but only for coverings that extend to coverings of the compactification $\bar{Y}$.

The authors themselves flag the central limitations. The Poincaré covering hypothesis, in their words, seems to be either a strong constraint or not easy to check; elementary examples show that the previous results are not valid for arbitrary Galois coverings of Zariski open sets (for instance, an $L^2$-vanishing statement is already false for coverings of punctured curves, and coverings with finite-order monodromy around divisor components are ramified at infinity, destroying positive injectivity radius of the pull-back metric); and the footnote to their Theorem 1.1.7 says of $AC^0$ that it holds in the Kähler case and should be true in general, without proof. A forthcoming article by the first named author is announced, meant to develop another model which allows the study of any Galois covering of a Zariski open subset of a compact Kähler manifold; no such article ever appeared. In dimension one, Jean (2022) proves a version of the $L^2$ index theorem for an infinite Galois covering of an open curve ramified at infinity in which the logarithmic formula acquires a Riemann–Hurwitz-type correction term.

Problem Statement

Construct an $L^2$-cohomology theory for (pullbacks of) harmonic Higgs bundles on arbitrary Galois coverings $p:X\to Y$ of Zariski open subsets $Y=\bar{Y}\setminus D$ of compact Kähler manifolds — that is, with no positive injectivity radius or bounded-geometry assumption on the pull-back of the Poincaré-type metric — and prove within it the analogues of the results established for Poincaré coverings by Dingoyan–Schumacher (2018):

(a) a finiteness statement: $\Gamma$-Fredholmness of the relevant Dolbeault/Higgs Laplacians — from which the Galois $\partial\bar\partial$-lemma is derived in the source — or an equivalent statement that the resulting $L^2$-cohomology groups have finite von Neumann dimension after the appropriate tensorization with $\mathcal{U}(\Gamma)$;

(b) an index/Euler-characteristic formula expressed by logarithmic characteristic integrals on $(\bar{Y},D)$, incorporating whatever correction terms the general case requires;

(c) degeneration at $E_2$ (or at the provably correct modified page) of the $L^2$ Dolbeault-to-de-Rham spectral sequence after tensorization with $\mathcal{U}(\Gamma)$ over $M(\Gamma)$;

(d) comparison of the analytic $L^2$ invariants with $L^2$ simplicial/equivariant cohomology, with homotopy invariance;

with the theory reducing to the known theorems when $p$ is a Poincaré covering. Because the verbatim statements are known to fail for arbitrary Galois coverings (coverings ramified at infinity being the basic obstruction), identifying the correct modified statements is an integral part of the problem. Note also that the comparison theorem of the source (its Theorem 1.1.7) is proved under the bundle hypothesis $AC^0$ of Definition 2.2.2, whose general validity is unknown — the paper's footnote to that theorem states only that it holds in the Kähler case and should be true in general — so obtaining (d) for arbitrary coverings, where that hypothesis has not been verified, is part of the problem. These four components are all required and constitute a single objective: an $L^2$ Hodge theory for harmonic Higgs bundles on coverings that is independent of bounded-geometry hypotheses.

References

1. Pascal Dingoyan, Georg Schumacher, Application of Cheeger-Gromov theory to the $l^2$-cohomology of harmonic Higgs bundles over covering of finite volume complete manifolds, preprint (2018), arXiv:1810.03863, DOI 10.48550/arXiv.1810.03863, https://arxiv.org/abs/1810.03863
2. Pascal Dingoyan, Some mixed Hodge structures on $l^2$-cohomology groups of coverings of Kähler manifolds, Mathematische Annalen 357 (2013), 1119–1174, DOI 10.1007/s00208-013-0909-2, https://doi.org/10.1007/s00208-013-0909-2 (preprint posted as arXiv:1006.2784 under the variant title 'Some mixed Hodge structure on $l^2$-cohomology of covering of Kähler manifolds', https://arxiv.org/abs/1006.2784)
3. Philippe Eyssidieux, Towards a $L^2$ cohomology theory for Hodge modules on infinite covering spaces: $L^2$ constructible cohomology and $L^2$ de Rham cohomology for coherent $\mathcal{D}$-modules, preprint (2022), arXiv:2203.06950, DOI 10.48550/arXiv.2203.06950, https://arxiv.org/abs/2203.06950
4. Bastien Jean, $L^2$-cohomology of a variation of Hodge structure for an infinite covering of an open curve ramified at infinity, preprint (2022), arXiv:2211.11605, DOI 10.48550/arXiv.2211.11605, https://arxiv.org/abs/2211.11605
5. Takuro Mochizuki, $L^2$-complexes and twistor complexes of tame harmonic bundles, preprint (2022), arXiv:2204.10443, DOI 10.48550/arXiv.2204.10443, https://arxiv.org/abs/2204.10443
6. Donu Arapura, Euler characteristics of Kollár-hyperbolic varieties, preprint (2025), arXiv:2509.04607, DOI 10.48550/arXiv.2509.04607, https://arxiv.org/abs/2509.04607