$L^2$-cohomology of wild harmonic bundles on quasi-compact Kähler manifolds: the surviving open core of the Jost–Yang–Zuo program

For a quasi-compact Kähler manifold $X=\bar{X}\setminus D$ ($\bar{X}$ compact Kähler, $D$ a simple normal crossing divisor), the Jost–Yang–Zuo program asks for a complete $L^2$-cohomology theory with coefficients in harmonic bundles, in the natural generality of such bundles: $L^2$-de Rham and $L^2$-Dolbeault/Higgs complexes, Poincaré-type lemmas giving fine resolutions, and a comparison theorem identifying these analytic complexes with a canonical topological or twistor-theoretic extension of the coefficients, together with finite-dimensionality, Hodge/twistor structure, and Kodaira-type vanishing. This program is now essentially complete for polarizable variations of Hodge structure (Zucker, Cattani–Kaplan–Schmid, Kashiwara–Kawai), for unipotent coefficients on curves (Jost–Yang–Zuo), and for all tame harmonic bundles on quasi-compact Kähler manifolds in every dimension (Mochizuki 2022), while the wild case is settled only in dimension one (Mochizuki, Astérisque 340). The precise nonempty open core is the corresponding theory for wild (irregular, non-tame) harmonic bundles on quasi-compact Kähler manifolds of dimension at least two, with the compactification not assumed to be projective.

Background

A quasi-compact Kähler manifold is a complex manifold $X$ obtained as the complement $X=\bar{X}\setminus D$ of a simple normal crossing divisor $D$ (a divisor locally defined, in suitable holomorphic coordinates, by the vanishing of a subset of those coordinates) in a compact Kähler manifold $\bar{X}$; it carries a complete Poincaré-type Kähler metric, i.e. one that near the divisor looks like a sum of terms $|z_k|^{-2}(\log|z_k|)^{-2}dz_k\,d\bar z_k$ in coordinates $z_k$ cutting out $D$. A local system $V$ on $X$, equivalently a linear representation $\rho:\pi_1(X)\to GL(n,\mathbb{C})$, gives a flat vector bundle. A harmonic metric on it is a Hermitian metric for which the associated $\rho$-equivariant map from the universal cover of $X$ to the symmetric space $GL(n,\mathbb{C})/U(n)$ is harmonic; by work of Donaldson, Corlette, and Simpson — the foundations of non-abelian Hodge theory, the dictionary between flat bundles and Higgs bundles on Kähler manifolds — such metrics exist uniquely for semisimple representations on compact Kähler manifolds. Following Simpson, a flat bundle with harmonic metric $h$ splits the flat connection into pieces that make the underlying smooth bundle a Higgs bundle $(E,\theta)$ — a holomorphic bundle $E$ with a holomorphic one-form $\theta\in H^0(\mathrm{End}(E)\otimes\Omega^1_X)$, $\theta\wedge\theta=0$, called the Higgs field — and the triple is called a harmonic bundle. On compact Kähler manifolds the de Rham cohomology of the flat bundle and the Higgs cohomology of $(E,\theta)$ are isomorphic by Kähler identities. On a non-compact $X$ these cohomologies are too large, and following Deligne and Zucker one replaces them by $L^2$-refinements: the $L^2$-de Rham complex of forms that are square-integrable with closed exterior derivative (measured with the Poincaré-type metric and $h$), and the $L^2$-Higgs and $L^2$-holomorphic Dolbeault complexes defined analogously. On the topological side one has the intersection cohomology of the direct image sheaf $j_*V$ ($j:X\hookrightarrow\bar{X}$), a middle-perversity refinement of sheaf cohomology insensitive to the analytic data. Zucker (1979) proved for a polarized variation of Hodge structure (VHS) — a flat bundle whose fibers carry a polarized Hodge structure varying holomorphically — on a punctured compact Riemann surface that the $L^2$-de Rham cohomology computes the intersection cohomology of $j_*V$; Cattani–Kaplan–Schmid and Kashiwara–Kawai (1987) established the higher-dimensional comparison for polarizable VHS on quasi-compact Kähler manifolds. Jost–Yang–Zuo launched a program extending this to harmonic bundles: they settled the VHS case on quasi-compact Kähler manifolds, proved the $L^2$-Poincaré and $L^2$-$\bar\partial$-Poincaré lemmas for unipotent harmonic bundles (i.e. with unipotent local monodromy, so the logarithms $N_k=\log\gamma_k$ of the loops around $D$ are nilpotent) on non-compact curves, and constructed harmonic metrics on unipotent bundles over quasi-compact Kähler manifolds of any dimension with asymptotics near $D$ controlled in the spirit of Schmid's Nilpotent Orbit Theorem (which describes the boundary asymptotics of a variation of Hodge structure in terms of nilpotent orbits); their survey explicitly poses the 'natural problem' of removing the unipotence assumption. Harmonic bundles in their natural generality need not be unipotent or even tame: following Mochizuki, a harmonic bundle on $X$ is tame if the closure of its spectral variety — the locus cut out in the cotangent bundle by $\det(\theta-\lambda\,\mathrm{id})=0$ — in the logarithmic cotangent bundle $\Omega^1_{\bar X}(\log D)$ is proper over $\bar{X}$, and wild otherwise; the wild case comprises irregular singularities, with pole growth of $\theta$ beyond logarithmic (tame) order. Mochizuki's theory of pure twistor D-modules (D-module-theoretic objects interpolating between flat and Higgs data through a twistor parameter $\lambda$, whose analytic $\mathcal R$-triple formulation is defined on general complex manifolds) supplies, for a tame harmonic bundle, a canonical 'minimal extension' analogous to the intersection complex. Building on this, Mochizuki (2022) proved for every tame harmonic bundle on $(\bar{X},D)$ with $\bar{X}$ compact Kähler that its $L^2$-complex is isomorphic in the derived category to the de Rham complex of the underlying pure twistor D-module, settling the tame case of the program in all dimensions; Deng–Hao (2024) independently constructed the fine $L^2$ resolution of the Dolbeault complex and Kodaira-type vanishing for tame harmonic bundles with nilpotent residues. For wild harmonic bundles the same comparison is known only in dimension one (Mochizuki, Astérisque 340), where the theory already shows that plain square-integrability must be modified — through a sectorwise decomposition of forms with respect to the Stokes structure, with polynomial-growth or rapid-decay conditions on the irregular components and $L^2$ conditions on the regular component, equivalently through algebraically determined sheaves defined via weight filtrations — and that the comparison target is a wild analogue of the intersection complex rather than the ordinary one; the monograph's prolongation machinery — asymptotic estimates and the canonical minimal-extension-type prolongation of good wild harmonic bundles — is analytic and local, developed on polydisks in arbitrary dimension, while projectivity enters only for the correspondence with algebraic semisimple holonomic D-modules. Existence of harmonic metrics with wild singularities is established in the projective setting, through Mochizuki's Kobayashi–Hitchin correspondence for good wild harmonic bundles (2021) — the equivalence between good wild harmonic bundles and polystable good filtered $\lambda$-flat bundles on smooth projective varieties — but no $L^2$-cohomology comparison in dimension at least two is known, even for projective compactifications.

Problem Statement

Let $\bar X$ be a compact Kähler manifold of dimension at least 2, $D$ a simple normal crossing divisor, and $X=\bar X\setminus D$ equipped with a Poincaré-type Kähler metric, with $\bar X$ not assumed projective. Let $(E,\bar\partial_E,\theta,h)$ be a wild harmonic bundle on $X$ that is good (i.e. lies in the class on which the associated wild pure twistor D-module is constructed via its prolongation to $\bar X$), namely a harmonic bundle that is not tame: the closure of its spectral variety $\Sigma_{(E,\theta)}$ in the logarithmic cotangent bundle $\Omega^1_{\bar X}(\log D)$ is not proper over $\bar X$. Construct the complete $L^2$-cohomology theory for such coefficients, mirroring what Zucker, Cattani–Kaplan–Schmid, and Kashiwara–Kawai did for variations of Hodge structure, Jost–Yang–Zuo did for unipotent coefficients — their Conjectures 1–2, the $L^2$-de Rham/intersection-cohomology comparison and the $L^2$-Dolbeault/$L^2$-Higgs quasi-isomorphism, are precisely the targets below — and Mochizuki did for tame harmonic bundles in all dimensions and for wild harmonic bundles on curves. Concretely: (i) define the sheaves of admissible $L^2$ forms and the resulting $L^2$-de Rham complex and $L^2$-holomorphic Dolbeault/$L^2$-Higgs complexes on $\bar X$, generalizing the modification of plain square-integrability established by the one-dimensional wild theory, and show that the resulting sheaf-level objects are independent of auxiliary choices; (ii) prove the $L^2$-Poincaré lemma and the $L^2$-$\bar\partial$-Poincaré lemma, i.e. that these complexes form fine resolutions of explicit sheaf-level prolongations of the flat and Higgs data on $\bar X$; and (iii) prove a comparison theorem identifying the $L^2$ complexes, in the derived category of sheaves on $\bar X$, with the de Rham complex of the wild pure twistor D-module associated to $(E,\bar\partial_E,\theta,h)$ with the minimal-extension normalisation — the wild analogue of the intersection complex of the local system — and hence identify the $L^2$-de Rham cohomology with the cohomology of that canonical target, which replaces the intersection cohomology of $j_*V$ in the non-unipotent setting. As in the VHS and tame cases, the structural consequences of such a comparison — finite-dimensionality of the resulting cohomology, its Hodge/twistor structure, and a Kodaira-type vanishing theorem — follow from the comparison theorem once it is established, as they do in the classical theory. A theory covering only curves or only tame bundles does not answer the problem.

References

1. Jürgen Jost, Yi-Hu Yang, Kang Zuo, 'Cohomologies of harmonic bundles on quasi-compact Kaehler manifolds' (2007), arXiv:0801.0194, https://arxiv.org/abs/0801.0194
2. Jürgen Jost, Yi-Hu Yang, Kang Zuo, 'Harmonic metrics on unipotent bundles over quasi-compact Kaehler manifolds' (2009), arXiv:0911.4759, https://arxiv.org/abs/0911.4759
3. Ya Deng, Feng Hao, 'Vanishing theorem for tame harmonic bundles via $L^2$-cohomology', Compositio Mathematica 160 (2024), no. 12, 2828–2855, doi:10.1112/s0010437x24007504; preprint arXiv:1912.02586 (2019), https://arxiv.org/abs/1912.02586
4. Takuro Mochizuki, '$L^2$-complexes and twistor complexes of tame harmonic bundles' (2022), arXiv:2204.10443, https://arxiv.org/abs/2204.10443
5. Takuro Mochizuki, 'Wild harmonic bundles and wild pure twistor D-modules', Astérisque 340, Société Mathématique de France (2011), doi:10.24033/ast.898, https://arxiv.org/abs/0803.1344
6. Takuro Mochizuki, 'Good Wild Harmonic Bundles and Good Filtered Higgs Bundles', Symmetry, Integrability and Geometry: Methods and Applications 17 (2021), doi:10.3842/sigma.2021.068, https://arxiv.org/abs/1902.08298
7. Jürgen Jost, Yi-Hu Yang, Kang Zuo, 'Cohomologies of unipotent harmonic bundles over noncompact curves', Journal für die reine und angewandte Mathematik 609 (2007), doi:10.1515/crelle.2007.062, https://doi.org/10.1515/crelle.2007.062
8. Jürgen Jost, Yi-Hu Yang, Kang Zuo, 'The cohomology of a variation of polarized Hodge structures over a quasi-compact Kähler manifold', Journal of Algebraic Geometry 16 (2007), no. 3, 401–434, doi:10.1090/s1056-3911-07-00468-7, https://doi.org/10.1090/s1056-3911-07-00468-7
9. Steven Zucker, 'Hodge theory with degenerating coefficients: $L^2$ cohomology in the Poincaré metric', Annals of Mathematics (2) 109 (1979), 415–476, doi:10.2307/1971221, https://doi.org/10.2307/1971221
10. Eduardo Cattani, Aroldo Kaplan, Wilfried Schmid, '$L^2$ and intersection cohomologies for a polarizable variation of Hodge structure', Inventiones Mathematicae 87 (1987), 217–252, doi:10.1007/bf01389415, https://doi.org/10.1007/bf01389415
11. Masaki Kashiwara, Takahiro Kawai, 'The Poincaré lemma for variations of polarized Hodge structure', Publications of the Research Institute for Mathematical Sciences 23 (1987), 345–407, doi:10.2977/prims/1195176545, https://doi.org/10.2977/prims/1195176545
12. Carlos T. Simpson, 'Harmonic bundles on noncompact curves', Journal of the American Mathematical Society 3 (1990), 713–770, doi:10.1090/s0894-0347-1990-1040197-8, https://doi.org/10.1090/s0894-0347-1990-1040197-8
13. Junyan Cao, Ya Deng, Christopher D. Hacon, Mihai Păun, 'Hodge theory for local systems and cohomological support loci' (2025), arXiv:2511.06773, https://arxiv.org/abs/2511.06773
14. Benoît Cadorel, Ya Deng, 'Picard hyperbolicity of manifolds admitting nilpotent harmonic bundles' (2021), arXiv:2107.07550, https://arxiv.org/abs/2107.07550