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Chuanjing Zhang

Publications and source records attributed to Chuanjing Zhang.

14 recordsLinked to original sources
arXiv2026

Non-abelian Hodge correspondence over singular Kähler spaces

In this paper, we establish the non-abelian Hodge correspondence over compact Kähler spaces with Kawamata log terminal (klt) singularities as well as over their regular loci, thereby extending the result of Greb-Kebekus-Peternell-Taji for projective klt varieties to the context of compact Kähler klt spaces. The proof relies on two key ingredients: first, we establish an equivalence over the regular loci-via harmonic bundles-between polystable Higgs bundles with vanishing orbifold Chern numbers and semi-simple flat bundles; second, we prove a descent theorem for semistable Higgs bundles with vanishing Chern classes along resolutions of singularities. As an application of our framework, we obtain a quasi-uniformization theorem for projective klt varieties with big canonical divisor that satisfy the orbifold Miyaoka-Yau equality.

math.DG↗
arXiv2026

Long-time behavior of the Hermitian-Yang-Mills flow on non-Kähler manifolds

In this paper, we study the long-time behavior of the Hermitian-Yang-Mills flow over compact Hermitian manifolds. We obtain the monotonicity of lower bound and upper bound of the eigenvalues of the mean curvature along the Hermitian-Yang-Mills flow. In the Gauduchon case, we show that the eigenvalues of the mean curvature converge to geometric invariants determined by the Harder-Narasimhan type. Furthermore, we generalize the Atiyah-Bott-Bando-Siu question to the non-Kähler case.

math.DG↗
arXiv2025

The Miyaoka-Yau inequality for minimal Kähler klt spaces

In this paper, we obtain the generalized Bogomolov inequality for reflexive Higgs sheaves defined on the regular locus of compact Kähler klt spaces. As an application, we establish the Miyaoka-Yau inequality for all minimal Kähler klt spaces. Apart from providing a self-contained formulation and investigation of Higgs sheaves on complex normal spaces, the analytical part of our approach is the establishment of $L^p$-approximate critical Hermitian structures for Higgs orbi-bundles on Gauduchon orbifolds. This also leads to the semistability (resp. generically nefness) of torsion-free sheaves under symmetric, exterior powers and tensor products in the singular setting.

math.DG↗
arXiv2024

Mean curvature positivity and rational connectedness

In this paper, we use Uhlenbeck-Yau's continuity method to establish the correspondence between the mean curvature positivity and the HN-positivity on holomorphic vector bundles over compact Hermitian manifolds. As its application, we get a differential geometric criterion for rational connectedness, i.e. we prove that a compact Kähler manifold is projective and rationally connected if and only if its holomorphic tangent bundle is mean curvature positive.

math.DG↗
arXiv2020

Projectively flat bundles and semi-stable Higgs bundles

The Corlette-Donaldson-Hitchin-Simpson's correspondence states that, on a compact Kähler manifold $(X, ω)$, there is a one-to-one correspondence between the moduli space of semisimple flat complex vector bundles and the moduli space of poly-stable Higgs bundles with vanishing Chern numbers. In this paper, we extend this correspondence to the projectively flat bundles case. We prove that there is an equivalence of categories between the category of $ω$-semi-stable (poly-stable) Higgs bundles $(E, \overline{\partial}_{E}, ϕ)$ with $(2rc_{2}(E)-(r-1)c_{1}^{2}(E))\cdot [ω]^{n-2}=0 $ and the category of (semi-simple) projectively flat bundles $(E, D)$ with $\sqrt{-1}F_{D}=α\otimes \mbox{Id}_{E}$ for some real (1,1)-form $α$. Furthermore, we also establish the above correspondence on some compact non-Kähler manifolds. As its application, we obtain a vanishing theorem of characteristic classes of projectively flat bundles.

math.DG↗
arXiv2019

Analytically stable Higgs bundles on some non-Kähler manifolds

In this paper, we study Higgs bundles on non-compact Hermitian manifolds. Under some assumptions for the underlying Hermitian manifolds which are not necessarily Kähler, we solve the Hermitian-Einstein equation on analytically stable Higgs bundles.

math.DG↗
arXiv2018

A Liouville theorem on complete non-Kähler manifolds

In this paper, we prove a Liouville theorem for holomorphic functions on a class of complete Gauduchon manifolds. This generalizes a result of Yau for complete Kähler manifolds to the complete non-Kähler case.

math.DG↗
arXiv2017

The limit of the Hermitian-Yang-Mills flow on reflexive sheaves

In this paper, we study the asymptotic behavior of the Hermitian-Yang-Mills flow on a reflexive sheaf. We prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the Harder-Narasimhan-Seshadri filtration, this answers a question by Bando and Siu.

math.DG↗
arXiv2016

A note on curvature estimate of the Hermitian-Yang-Mills flow

In this paper, we study the curvature estimate of the Hermitian-Yang-Mills flow on holomorphic vector bundles. In one simple case, we show that the curvature of the evolved Hermitian metric is uniformly bounded away from the analytic subvariety determined by the Harder-Narasimhan-Seshadri filtration of the holomorphic vector bundle.

math.DG↗
arXiv2016

The conical complex Monge-Ampère equations on Kähler manifolds

In this paper, by providing the uniform gradient estimates for a sequence of the approximating equations, we prove the existence, uniqueness and regularity of the conical parabolic complex Monge-Ampère equation with weak initial data. As an application, we prove a regularity estimates, that is, any $L^{\infty}$-solution of the conical complex Monge-Ampère equation admits the $C^{2,α,β}$-regularity.

math.AP↗
arXiv2016

Semi-stable Higgs sheaves and Bogomolov type inequality

In this paper, we study semistable Higgs sheaves over compact Kähler manifolds, we prove that there is an approximate admissible Hermitian-Einstein structure on a semi-stable reflexive Higgs sheaf and consequently, the Bogomolove type inequality holds on a semi-stable reflexive Higgs sheaf.

math.DG↗