Admissibility of the limiting Hermitian–Einstein metric for reflexive Higgs sheaves on compact Kähler klt spaces: uniform Higgs-field bounds along the orbifold Hermitian–Yang–Mills flow (Question 3.6 of Zhang–Zhang–Zhang)

The non-abelian Hodge correspondence over compact Kähler spaces with Kawamata log terminal (klt) singularities, constructed by Zhang–Zhang–Zhang (arXiv:2601.13071), produces pluri-harmonic metrics as local limits of a Hermitian–Yang–Mills flow on an orbifold modification, but leaves open whether the limiting Hermitian–Einstein metric is admissible — the primary obstacle being a uniform bound on the Higgs-field norm along the flow, posed explicitly as Question 3.6. The question remains open: the existing admissible Hermitian–Yang–Mills literature covers only the Higgs-field-free case of reflexive sheaves, and no known work establishes this estimate in the Higgs-field case. The problem is Question 3.6: prove the uniform estimate, or exhibit an explicit space and sheaf for which the Higgs-field norm $\|\theta_{X^{\mathrm{reg}}}\|_{H(t),\omega}$ is unbounded along the flow.

Background

On a compact Kähler manifold, the non-abelian Hodge correspondence (Corlette, Donaldson, Simpson) sets up an equivalence between three kinds of objects: semisimple flat bundles (equivalently, semisimple complex representations of the fundamental group); polystable Higgs bundles with vanishing Chern classes; and harmonic bundles. A Higgs bundle is a pair $(E,\theta)$ of a holomorphic vector bundle $E$ and a Higgs field $\theta\in\Omega_X^1\otimes\mathrm{End}(E)$, an operator $\theta\colon E\to E\otimes\Omega_X^1$ satisfying $\theta\wedge\theta=0$; it is (poly)stable in the sense of slope stability with the Higgs field included: every proper coherent subsheaf invariant under the Higgs field has strictly smaller normalized degree (degree divided by rank), and polystable means a direct sum of stable summands of equal slope. A harmonic bundle is a Higgs bundle together with a Hermitian metric $H$ such that the associated connection $D=D_H+\theta+\theta^{*H}$ (Chern connection plus the Higgs field and its metric adjoint) is flat; such a metric is called pluri-harmonic. The analytic input is the Hitchin–Kobayashi or Donaldson–Uhlenbeck–Yau (DUY) theorem: a holomorphic bundle over a compact Kähler manifold is polystable if and only if it admits a Hermitian–Einstein metric, that is, a metric $H$ for which the contraction of the Chern curvature $\sqrt{-1}\Lambda_\omega F_H$ equals a scalar multiple of the identity. Simpson turned these PDE existence results into the correspondence and its uniformization applications.

For singular spaces the correspondence was first extended by Greb–Kebekus–Peternell–Taji to projective varieties with Kawamata log terminal (klt) singularities — the standard class of mild singularities in birational geometry, characterized by divisors of discrepancies greater than $-1$ over the singular locus — where one works with reflexive Higgs sheaves on the regular locus $X^{\mathrm{reg}}$ (the smooth part of $X$) - sheaves equal to their double dual, hence determined by their restriction to any dense open set, the standard substitute for vector bundles on a singular space - and characterizes them by pullbacks to resolutions and quasi-étale covers (finite covers that are étale in codimension one). Zhang–Zhang–Zhang (arXiv:2601.13071) extended this to all compact Kähler klt spaces. Their strategy adapts the Bando–Siu technique: on an orbifold modification $f\colon Y\to X$ — a bimeromorphic map from a complex space with only quotient singularities, with the indeterminacy locus of $f^{-1}$ of codimension at least $3$, as constructed by Kollár–Ou — one runs a Hermitian–Yang–Mills (HYM) flow $H(t)$ on the pulled-back reflexive Higgs sheaf and takes a limit. Their Theorem 3.5 gives convergence $H(t)\to H_\infty$ in $C^\infty_{\mathrm{loc}}$ away from the singular locus $\Sigma$, with $\sqrt{-1}\Lambda_\omega(F_{H_\infty},\theta)=\lambda\,\mathrm{Id}$ there; the vanishing of the orbifold Chern numbers (Chern classes defined via the orbifold modification, generalizing the usual Chern–Weil theory to quotient-type singularities) forces the limit to be pluri-harmonic, and the non-abelian Hodge correspondence follows. Crucially, the authors circumvent any a priori control of the limit metric near $\Sigma$: they characterize the correspondence without imposing admissibility assumptions on the pluri-harmonic metrics.

In the parallel 'admissible metrics' framework for singular spaces, one asks more: a Hermitian metric on a reflexive sheaf over the regular locus is admissible when it extends across the singular locus with the $L^2$-type bounds on curvature and its derivatives that make Chern–Weil computations and $L^2$-arguments legitimate on the singular space. In that framework, Ou studied admissible metrics conditional on a uniform Sobolev inequality; Chen proved a complete DUY theorem over normal varieties via admissible HYM connections; Cao–Graf–Naumann–Paun–Peternell–Wu constructed Hermitian–Einstein metrics in related singular settings; and Jinnouchi obtained admissible HYM metrics for slope-stable reflexive sheaves on klt Kähler–Einstein varieties. All of these treat ordinary reflexive sheaves, with no Higgs field. For Higgs sheaves on compact Kähler klt spaces, the uniform estimate of the Higgs-field norm along the HYM flow — known in the smooth manifold case by the work of Li–Zhang–Zhang — is explicitly left open in arXiv:2601.13071, and its absence is stated there to be the primary obstacle to the admissibility of $H_\infty$ and hence to a DUY theorem in this singular Higgs context.

Problem Statement

Let $X$ be a compact Kähler space with Kawamata log terminal (klt) singularities of dimension $n$, let $\omega_0,\dots,\omega_{n-2}$ be Kähler forms, let $\omega$ be the smooth Hermitian metric on $X$ determined by $\omega^{n-1}=\omega_0\wedge\cdots\wedge\omega_{n-2}$ (equation (3.10) of arXiv:2601.13071), and let $(E_{X^{\mathrm{reg}}},\theta_{X^{\mathrm{reg}}})$ be a reflexive Higgs sheaf on the regular locus of $X$ that is polystable with respect to $(\omega_0,\dots,\omega_{n-2})$ with vanishing orbifold Chern numbers. Following Section 3 of Zhang–Zhang–Zhang (arXiv:2601.13071), run the Hermitian–Yang–Mills flow $H(t)$ for $(E_{X^{\mathrm{reg}}},\theta_{X^{\mathrm{reg}}})$ induced on an orbifold modification of $X$; by their Theorems 3.5 and 3.2, a subsequence converges in $C^\infty_{\mathrm{loc}}$ on $X\setminus\Sigma$ to a limiting Hermitian–Einstein metric $H_\infty$, which is pluri-harmonic.

Question (Question 3.6 of arXiv:2601.13071): for $t_0>0$, is the Higgs-field norm $\|\theta_{X^{\mathrm{reg}}}\|_{H(t),\omega}$ uniformly bounded for all $t>t_0$ along the flow?

The scientific objective of which this estimate is the stated primary obstacle is the admissibility of the limiting metric: prove that $H_\infty$ is admissible in the sense of the admissible-metrics framework for singular spaces (extendable across the singular locus with the $L^2$-type curvature bounds under which Chern–Weil theory and $L^2$-arguments are legitimate on $X$), thereby establishing the Donaldson–Uhlenbeck–Yau theorem for reflexive Higgs sheaves on compact Kähler klt spaces in the admissible sense — the analytic underpinning that the non-abelian Hodge correspondence of arXiv:2601.13071 deliberately avoids assuming. The uniform bound (Question 3.6) is the precise open sub-question of that objective: prove that the Higgs-field norm $\|\theta_{X^{\mathrm{reg}}}\|_{H(t),\omega}$ is uniformly bounded for all $t>t_0$ along the flow, or exhibit an explicit such space and sheaf for which it is unbounded. The admissibility of $H_\infty$ and the admissible Donaldson–Uhlenbeck–Yau statement are the broader context in which this estimate sits, in the natural generality of the source (all compact Kähler klt spaces and all polystable reflexive Higgs sheaves with vanishing orbifold Chern numbers).

References

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