Extending the twistor-complex–$L^2$-complex comparison for harmonic bundles to general singular supports and to the wild case in higher dimensions

For a tame harmonic bundle on the complement of a simple normal crossing hypersurface in a compact Kähler manifold, Mochizuki (arXiv:2204.10443, 2022) proved a comparison isomorphism in the derived category between the twistor complex — the de Rham complex of the $\mathcal R_X$-module underlying the associated pure twistor $\mathcal D$-module — and the $L^2$-complex defined by the flat $\lambda$-connection and a Poincaré-like Kähler metric, together with its specializations at each $\lambda$. The problem, drawn from the paper's own delimitation of its scope, is to extend this comparison in two directions: (i) to tame harmonic bundles whose singular support is more general than a simple normal crossing divisor, where only polarized variations of Hodge structure and 0-tame harmonic bundles are covered by Shentu–Zhao; and (ii) to wild (irregular) harmonic bundles on complex manifolds of dimension at least two, where only the one-dimensional case is known from Mochizuki's theory of wild harmonic bundles and wild pure twistor $\mathcal D$-modules. No published work establishing either extension is known.

Background

Harmonic bundles are the central objects of nonabelian Hodge theory, the correspondence on complex manifolds between flat bundles and Higgs bundles mediated by harmonic metrics. A harmonic bundle on a complex manifold $Y$ consists of a holomorphic vector bundle $(E,\bar\partial_E)$, a Higgs field $\theta$ (a holomorphic section of $\mathrm{End}(E)\otimes\Omega_Y^1$ satisfying $\theta\wedge\theta=0$), and a Hermitian metric $h$ such that the connection $D_1=\nabla^h+\theta+\theta^\dagger$ is flat, where $\nabla^h$ is the Chern connection and $\theta^\dagger$ the adjoint of $\theta$; harmonic bundles on punctured curves were introduced and classified by Simpson in this framework. A polarized variation of Hodge structure (a holomorphically varying family of Hodge structures with a flat connection satisfying Griffiths transversality, i.e. the connection maps each Hodge summand only to itself and the adjacent one) produces a harmonic bundle via its Hodge decomposition, and harmonic bundles interpolate between flat bundles and Higgs bundles.

For a harmonic bundle $(E,\bar\partial_E,\theta,h)$, Simpson's twistor construction produces a holomorphic vector bundle $\mathcal E$ on $\mathbb C_\lambda\times Y$ with a flat $\lambda$-connection $D=\lambda\partial_{E,h}+\theta+\bar\partial_E+\lambda\theta^\dagger$ (where $\partial_{E,h}$ is the $(1,0)$-part of the Chern connection), deforming between the Higgs bundle at $\lambda=0$ and the flat bundle at $\lambda=1$.

Fix a compact Kähler manifold $X$ (a compact complex manifold whose Hermitian metric has closed associated $(1,1)$-form) and a simple normal crossing (SNC) hypersurface $H$ (locally a union of coordinate hyperplanes meeting transversely). A harmonic bundle on $X\setminus H$ is called tame if the closure of its spectral variety (the support of the coherent sheaf induced by $\theta$ on the cotangent bundle) is proper over $X$ inside the logarithmic cotangent bundle; tame harmonic bundles constitute the regular-singular case, while wild (irregular) harmonic bundles — whose flat connections have irregular singularities along the boundary, governed in dimension one by Stokes data, the asymptotic data classifying irregular singular connections — lie outside this class.

Two natural complexes of sheaves on $X$ extend the twistor de Rham complex from $X\setminus H$. The first is algebro-geometric: the theory of pure twistor $\mathcal D$-modules (the twistor-theoretic generalization of Saito's pure Hodge modules — the category extending polarizable variations of Hodge structure to singular spaces — developed by Sabbah and Mochizuki, with the mixed theory in Mochizuki's book on mixed twistor $\mathcal D$-modules) attaches to a tame harmonic bundle a pure twistor $\mathcal D$-module whose underlying $\mathcal R_X$-module $\mathcal E$ (a Rees-type deformation over the $\lambda$-line of a filtered $\mathcal D$-module) yields the twistor complex $\mathrm{DR}(\mathcal E)$; in the Hodge case this is the de Rham complex of the minimal extension (the intermediate extension across the singular locus, which admits no sub- or quotient-object supported there), i.e. a resolution of the intersection complex (the complex computing intersection cohomology, a Poincaré-duality cohomology theory for singular spaces). The second is analytic: the $L^2$-complex $C^\bullet_{L^2}(E,D,h)$ of locally $L^2$ sections $\tau$ such that $\tau$ and $D\tau$ are square-integrable with respect to $h$ and a Poincaré-like Kähler metric on $X\setminus H$ (a metric comparable to $\sum_i|z_i|^{-2}(-\log|z_i|^2)^{-2}dz_i\,d\bar z_i$ near $H$).

The comparison question asks whether these two complexes are isomorphic in the derived category. Zucker (1979) proved this for polarized variations of Hodge structure on punctured curves, identifying $L^2$-cohomology with intersection cohomology; Cattani–Kaplan–Schmid and Kashiwara–Kawai (1987) established the higher-dimensional comparison in the Hodge case with SNC singularities. Mochizuki (2007) and Sabbah (2005) handled tame harmonic bundles on punctured curves within the twistor $\mathcal D$-module framework, and Mochizuki (Astérisque 340, 2011) settled the wild case in dimension one in the framework of wild pure twistor $\mathcal D$-modules. Mochizuki (2022) then established the full tame SNC comparison in arbitrary dimension: natural isomorphisms $\mathrm{DR}(\mathcal E)\simeq C^\bullet_{L^2}(E,D,h)$ and, for each specialization $\lambda$, $C^\bullet_{\mathrm{tw}}(\mathcal E,\lambda)\simeq C^\bullet_{L^2}(\mathcal E^\lambda,D^\lambda,h)$, restricting to the natural quasi-isomorphism over $X\setminus H$. As an application he proved the Hard Lefschetz theorem (the isomorphisms induced by cupping with a Kähler class in complementary degrees) for push-forwards of regular polarized pure twistor $\mathcal D$-modules under proper, possibly non-projective, Kähler morphisms, extending M. Saito's decomposition theorem (the splitting of such push-forwards into direct sums of shifted simple pieces) for projective morphisms of Kähler manifolds.

What lies outside the 2022 paper — as its introduction and Remark 1.4 make explicit — is the extension of this comparison to singular supports more general than SNC divisors, and to wild harmonic bundles in dimension at least two. In the first direction, Shentu–Zhao developed the $L^2$-theory for polarized variations of Hodge structure and for 0-tame harmonic bundles — those whose multivalued flat sections satisfy, for every $\epsilon>0$ and any local generators $f_1,\dots,f_r$ of the ideal of $Z_1$, the two-sided growth estimate $(\sum_{i}|f_i|^2)^{\epsilon}\lesssim|v|_h\lesssim(\sum_{i}|f_i|^2)^{-\epsilon}$ (Shentu–Zhao, arXiv:2103.04030) — on general closed analytic pairs $(Z_0,Z_1)$, using a suitable distinguished complete Kähler metric on $Z_0\setminus Z_1$; the general tame case, and the identification of the twistor-theoretic structures, remained open. In the second direction, no extension of the comparison to wild harmonic bundles in dimension $\ge 2$ is known.

Problem Statement

Let $X$ be a compact Kähler manifold and $H\subset X$ a simple normal crossing hypersurface. Mochizuki (2022, Theorems 1.2 and 1.3) proved that for any tame harmonic bundle $(E,\bar\partial_E,\theta,h)$ on $X\setminus H$ there are natural isomorphisms in the derived category of $(\mathcal O_{\mathbb C_\lambda})_X$-modules

$$\mathrm{DR}(\mathcal E)\simeq C^\bullet_{L^2}(E,D,h)$$

between the twistor complex (the de Rham complex of the $\mathcal R_X$-module $\mathcal E$ underlying the pure twistor $\mathcal D$-module associated with the harmonic bundle) and the $L^2$-complex with respect to $h$ and a Poincaré-like Kähler metric on $X\setminus H$, where $D=\lambda\partial_{E,h}+\theta+\bar\partial_E+\lambda\theta^\dagger$ and $\partial_{E,h}$ is the $(1,0)$-part of the Chern connection, together with the analogous isomorphisms for the specializations

$$C^\bullet_{\mathrm{tw}}(\mathcal E,\lambda)\simeq C^\bullet_{L^2}(\mathcal E^\lambda,D^\lambda,h)\quad(\lambda\in\mathbb C),$$

where $C^\bullet_{\mathrm{tw}}(\mathcal E,\lambda)$ is the complex on $X$ related to the specialization of the $\mathcal R_X$-module $\mathcal E$ to $\{\lambda\}\times X$, and $(\mathcal E^\lambda,D^\lambda)$ is the holomorphic bundle $(E,\bar\partial_E+\lambda\theta^\dagger)$ on $X\setminus H$ with the flat $\lambda$-connection $D^\lambda=\lambda\partial_{E,h}+\theta+\bar\partial_E+\lambda\theta^\dagger$; both isomorphisms restrict to the natural quasi-isomorphism over $X\setminus H$.

Extend this comparison isomorphism to the following settings, in the natural generality in which the objects are defined:

(i) Tame harmonic bundles on $Z_0\setminus Z_1$ — tameness being understood in the same spectral-variety-properness sense as above, taken with respect to a log resolution of the pair $(Z_0,Z_1)$ that renders the preimage of $Z_1$ a simple normal crossing divisor — where $Z_1\subset Z_0$ are closed complex analytic subsets of a compact Kähler manifold, $Z_0\setminus Z_1$ is a locally closed complex submanifold, and $Z_1$ is not assumed to be (contained in) a simple normal crossing hypersurface — including the identification of the induced pure twistor $\mathcal D$-module structure, not only the underlying intersection-type complex. The case of polarized variations of Hodge structure and of 0-tame harmonic bundles is known (Shentu–Zhao); the target is the general tame case.

(ii) Wild (irregular, non-tame) harmonic bundles on complex manifolds of dimension at least $2$, whose flat connections have irregular singularities along the boundary divisor — the one-dimensional case being known from Mochizuki's theory of wild harmonic bundles and wild pure twistor $\mathcal D$-modules (Astérisque 340).

The two extension directions are independent, neither subsuming the other; the source paper's own delimitation of its coverage identifies both directions as lying outside the established comparison theory.

References

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