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Xuemiao Chen

Publications and source records attributed to Xuemiao Chen.

12 recordsLinked to original sources
arXiv2026

Point Singularities and Local Third Chern Classes for Rank-Two Torsion-free Sheaves on Threefolds

In this paper, motivated by singularity formation in gauge theory, we study the local third Chern class contribution carried by isolated point singularities of rank-two torsion-free sheaves on complex threefolds. In the local rank-two setting considered here, the invariant is defined in terms of finite-length local algebraic data at the singular point. We prove that it can be computed from data on the total family; in particular, it is deformation invariant. We also prove that its parity recovers a topological invariant of the underlying smooth complex rank-two vector bundle on the boundary sphere. We then give a relative K-theoretic interpretation: a self-dual complex naturally associated with the sheaf defines a local $K$-theoretic charge, and this charge is equal to the local third Chern class. For rank-two reflexive sheaves, we relate the same invariant to several classical algebraic quantities, including the Fitting scheme and the Buchsbaum-Rim multiplicity. We also discuss applications to the boundary of moduli spaces of Hermitian-Yang-Mills connections.

math.DG↗
arXiv2026

Point Singularities and Bubbling in Degenerations of Rank-Two Bundles on Threefolds

We study one-parameter degenerations of rank-two vector bundles on complex threefolds to a rank-two torsion-free sheaf with an isolated point singularity. We prove a rigidity identity: the algebraic bubbling multiplicity of the central fiber equals one half of the Ext-length of the singularity of its reflexive hull. Furthermore, bubbling is forced when a family develops such an isolated point singularity. We use this identity to obtain smoothability obstructions and construct sharp local smoothings. We realize the local example from earlier joint work with Sun as a global degeneration of smooth Hermitian-Yang-Mills connections. Rescaling this degeneration produces a smooth non-flat Hermitian-Yang-Mills connection on $\mathbb C^3$ with density one at infinity, whose tangent cone at infinity has flat connection part and a multiplicity-one line as the blow-up cycle. We also construct smoothings of elementary modifications of projective-cone singularities with explicit algebraic bubbles. These examples give local models for Hermitian-Yang-Mills point bubbling in complex dimension three and distinguish this phenomenon from bubbling along complex codimension-two loci.

math.DG↗
arXiv2025

Bubbling of rank two bundles over surfaces

In this paper, motivated by the singularity formation of ASD connections in gauge theory, we study an algebraic analogue of the singularity formation of families of rank two holomorphic vector bundles over surfaces. For this, we define a notion of fertile families bearing bubbles and give a characterization of it using the related discriminant. Then we study families that locally form the singularity of the type $\mathcal{O}\oplus \mathcal{I}$ where $\mathcal{I}$ is an ideal sheaf defining points with multiplicities. We prove the existence of fertile families bearing bubbles by using elementary modifications of the original family. As applications, we study bubble trees for a few families that form singularities of low multiplicities and use examples to give negative answers to some plausible general questions.

math.DG↗
arXiv2025

Admissible Hermitian-Yang-Mills connections over normal varieties

In this paper, we first prove a complete version of the Donaldson-Uhlenbeck-Yau theorem over normal varieties, including normal Kaehler varieties and projective normal varieties with multiple polarizations. In particular, this gives the polystability of reflexive sheaves under symmetric and exterior powers and tensor products. As a consequence of the singular Donaldson-Uhlenbeck-Yau theorem, the complete Hitchin-Kobayashi correspondence over normal varieties smooth in codimension two is built by showing that an admissible Hermitian-Yang- Mills connection defines a polystable reflexive sheaf. Furthermore, it is shown that the Hermitian-Yang-Mills connection gives a lower bound for the discriminants of any Kaehler resolutions, which gives a Bogomolov-Gieseker inequality over normal varieties and a characterization of the equality using projectively flat connections. We discuss typical cases including normal surfaces and varieties smooth in codimension two where we could simplify the Bogomolov-Gieseker inequality and endow it with topological meanings. We also prove the Bogomolov-Gieseker inequality for semistable reflexive sheaves and characterize the class of semistable sheaves that satisfy the Bogomolov-Gieseker equality. Finally, as another application, we give a new criteria for when a normal Kaehler variety with trivial first Chern class is a finite quotient of torus.

math.DG↗
arXiv2025

On Vafa-Witten equations over Kaehler manifolds

In this paper, we study the analytic properties of solutions to the Vafa-Witten equation over a compact Kaehler manifold. Simple obstructions to the existence of nontrivial solutions are identified. The gauge theoretical compactness for the $\mathbb{C}^*$ invariant locus of the moduli space is shown to behave similarly as the Hermitian-Yang-Mills connections. More generally, this holds for solutions with uniformly bounded spectral covers such as nilpotent solutions. When spectral covers are unbounded, we manage to take limits of the renormalized Higgs fields which are intrinsically characterized by the convergence of the associated spectral covers. This gives a simpler proof for Taubes' results on rank two solutions over Kaehler surfaces together with a new complex geometric interpretation. The moduli space of $SU(2)$ monopoles and some related examples are also discussed in the final section.

math.DG↗
arXiv2022

A Donaldson-Uhlenbeck-Yau theorem for normal varieties and semistable bundles on degenerating families

In this paper, we first prove a Donaldson-Uhlenbeck-Yau theorem over projective normal varieties smooth in codimension two. As a consequence we deduce the polystability of (dual) tensor products of stable reflexive sheaves, and we give a new proof of the Bogomolov-Gieseker inequality, along with a precise characterization of the case of equality. This also improves several previously known algebro-geometric results on normalized tautological classes. We study the limiting behavior of semistable bundles over a degenerating family of projective normal varieties. In the case of a family of stable vector bundles, we study the degeneration of the corresponding HYM connections and these can be characterized from the algebro-geometric perspective. In particular, this proves another version of the singular Donaldson-Uhlenbeck-Yau theorem for the normal projective varieties in the central fiber. As an application, we apply the results to the degeneration of stable bundles through the deformation to projective cones, and we explain how our results are related to the Mehta-Ramanathan restriction theorem.

math.AG↗
arXiv2021

Compactness for $Ω$-Yang-Mills connections

On a Riemannian manifold of dimension $n$ we extend the known analytic results on Yang-Mills connections to the class of connections called $Ω$-Yang-Mills connections, where $Ω$ is a smooth, not necessarily closed, $(n-4)$-form. Special cases include $Ω$-anti-self-dual connections and Hermitian-Yang-Mills connections over general complex manifolds. By a key observation, a weak compactness result is obtained for moduli space of smooth $Ω$-Yang-Mills connections with uniformly $L^2$ bounded curvature, and it can be improved in the case of Hermitian-Yang-Mills connections over general complex manifolds. A removable singularity theorem for singular $Ω$-Yang-Mills connections on a trivial bundle with small energy concentration is also proven. As an application, it is shown how to compactify the moduli space of smooth Hermitian-Yang-Mills connections on unitary bundles over a class of balanced manifolds of Hodge-Riemann type. This class includes the metrics coming from multipolarizations, and in particular, the Kaehler metrics. In the case of multipolarizations on a projective algebraic manifold, the compactification of smooth irreducible Hermitian-Yang-Mills connections with fixed determinant modulo gauge transformations inherits a complex structure from algebro-geometric considerations.

math.DG↗
arXiv2021

The nonabelian Hodge correspondence for balanced hermitian metrics of Hodge-Riemann type

This paper extends the nonabelian Hodge correspondence for Kaehler manifolds to a larger class of hermitian metrics on complex manifolds called balanced of Hodge-Riemann type. Essentially, it grows out of a few key observations so that the known results, especially the Donaldson-Uhlenbeck-Yau theorem and Corlette's theorem, can be applied in our setting. Though not necessarily Kaehler, we show that the Sampson-Siu Theorem proving that harmonic maps are pluriharmonic remains valid for a slightly smaller class by using the known argument. Special important examples include those balanced metrics arising from multipolarizations.

math.DG↗
arXiv2020

Singularities of Hermitian-Yang-Mills connections and Harder-Narasimhan-Seshadri filtrations

This is the first of a series of papers where we relate tangent cones of Hermitian-Yang-Mills connections at an isolated singularity to the complex algebraic geometry of the underlying reflexive sheaf, when the sheaf is locally modelled on the pull-back of a holomorphic vector bundle from the projective space. In this paper we shall impose an extra assumption that the graded sheaf determined by the Harder-Narasimhan-Seshadri filtrations of the vector bundle is reflexive. In general we conjecture that the tangent cone is uniquely determined by the double dual of the associated graded object of a Harder-Narasimhan-Seshadri filtration of an algebraic tangent cone, which is a certain torsion-free sheaf on the projective space. In this paper we also prove this conjecture when there is an algebraic tangent cone which is locally free and stable.

math.DG↗
arXiv2020

Analytic tangent cones of admissible Hermitian-Yang-Mills connections

In this paper we study the analytic tangent cones of admissible Hermitian-Yang-Mills connections near a homogeneous singularity of a reflexive sheaf, and relate it to the Harder-Narasimhan-Seshadri filtration. We also give an algebro-geometric characterization of the bubbling set. This strengthens our previous result.

math.DG↗
arXiv2018

Algebraic tangent cones of reflexive sheaves

We study the notion of algebraic tangent cones at singularities of reflexive sheaves. These correspond to extensions of reflexive sheaves across a negative divisor. We show the existence of optimal extensions in a constructive manner, and we prove the uniqueness in a suitable sense. The results here are an algebro-geometric counterpart of our previous study on singularities of Hermitian-Yang-Mills connections.

math.DG↗