arXiv subjects

Nianzi Li

Publications and source records attributed to Nianzi Li.

4 recordsLinked to original sources

arXiv2026

Nonlinear Hodge correspondence for morphisms

We study Higgs sections, flat sections, and their higher-dimensional and functorial analogues in the setting of nonlinear harmonic bundles. For a harmonic vector bundle over a compact K\"ahler manifold, a global section is flat if and only if it is a Higgs section. We show that for general nonlinear harmonic bundles this equivalence requires the vanishing of a degree obstruction. We then extend the result to sub-fibrations and to morphisms by interpreting morphisms as graph sub-fibrations in a fiber product.

math.DG↗

arXiv2026

Connections, metrics and Higgs fields on complex fiber bundles

We give a representation of the extension class associated to a holomorphic fibration by curvature, generalizing the work of Atiyah on holomorphic principal bundles in a natural way. As an application, we obtain a nonlinear analogue of the classical result of Weil on characterizing the existence of flat connections on holomorphic vector bundles over compact Riemann surfaces. We further establish a faithful functor from the category of nonlinear flat bundles reductive of K\"ahler type to the category of nonlinear Higgs bundles over the same base, which is assumed to be a compact complex manifold of K\"ahler type. Finally, we establish a notion of nonlinear harmonic bundle and prove that the variation of nonabelian Hodge structure is a nonlinear harmonic bundle in the rank one case and in the semisimple case.

math.DG↗

arXiv2025

The asymptotics of the $\mathrm{SL}_2(\mathbb{C})$-Hitchin metric on the singular locus: subintegrable systems

We study the asymptotic hyperk\"ahler geometry of the $\mathrm{SL}_2(\mathbb{C})$-Hitchin moduli space over the singular fibers of the Hitchin fibration. We extend the previously known exponential convergence results for solutions to the Hitchin equation to the class of locally fiducial Higgs bundles defined by a special local description at the singularities of the spectral curve. This condition is satisfied by the Higgs bundles contained in certain subintegrable systems introduced by Hitchin. We prove that the restriction of the hyperk\"ahler metric to the subintegrable system converges exponentially fast to the corresponding semi-flat metric along a ray $(\mathcal{E},t\varphi)$. This answers a question posed by Hitchin in \cite{Hitchin2021subintegrable_special_Kaehler}. More generally, we prove that for each stratum of quadratic differentials there is a closed subset of the corresponding Hitchin fibers, such that the restricted hyperk\"ahler metric converges to a generalized semi-flat metric.

math.DG↗

arXiv2024

Asymptotic Geometry of the Moduli Space of Rank Two Irregular Higgs Bundles over the Projective Line

We study the asymptotic behavior of Hitchin's hyperkähler metric on the moduli space of rank two irregular Higgs bundles over $\mathbb{C}P^1$. Along a generic curve, we prove that the Hitchin metric is asymptotic to the semiflat metric at an arbitrary polynomial order. When there are no weakly parabolic singularities, the rate is exponential. In the case of four-dimensional moduli spaces, we prove that the semiflat metric is asymptotic to an ALG/ALG$^\ast$ model metric.

math.DG↗