arXiv subjects

Runhong Zong

Publications and source records attributed to Runhong Zong.

10 recordsLinked to original sources

arXiv2025

Deformation theory of parabolic representation pairs

In this paper, we introduce the notions of parabolic representation pairs and the parabolic representation pair variety. We investigate the deformation theory of parabolic representation pairs. The Zariski tangent space and the tangent quadratic cone of the parabolic representation pair variety are described. By the Riemann--Hilbert--Deligne correspondence, we pro-represent the analytic germs of parabolic representation pair variety by functors related to certain groupoids of parabolic logarithmic flat bundles. Under suitable assumptions, we prove that the differential graded Lie algebra (DGLA) controlling the deformation of a parabolic logarithmic flat bundle is partially formal. This leads to the quadraticity of the subvariety of parabolic representation pair variety, which consists of parabolic representation pairs with fixed eigenvalues of monodromies, at certain generic points. Finally, we construct the moduli space of weighted parabolic representation pairs, and, by means of quiver representation theory, we establish the Kobayashi--Hitchin-type theorem for polystable parabolic representation pairs.

math.AG↗

arXiv2024

Parahoric reduction theory of formal connections (or Higgs fields)

In this paper, we establish the parahoric reduction theory of formal connections (or Higgs fields) on a formal principal bundle with parahoric structures, which generalizes Babbitt-Varadarajan's result for the case without parahoric structures [5] and Boalch's result for the case of regular singularity [9]. As applications, we prove the equivalence between extrinsic definition and intrinsic definition of regular singularity and provide a criterion of relative regularity for formal connections, and also demonstrate a parahoric version of Frenkel-Zhu's Borel reduction theorem of formal connections [23].

math.AG↗

arXiv2023

Universal Constants, Law of Inertia and Emergent Geometry

In this paper, we only treat the law of inertia as the first principle, then a nontrivial geometry emerges by introducing more universal constants, in which the main ideas appearing in deformed special relativity (DSR), (Anti-)de Sitter special relativity [(A)dSSR] and bimetric gravity (BMG) have been contexture.

math-ph↗

arXiv2023

Topological Structures of Moduli Spaces of Curves and Anabelian Geometry in Positive Characteristic

In the present paper, we study a new kind of anabelian phenomenon concerning the smooth pointed stable curves in positive characteristic. It shows that the topological structures of moduli spaces of curves can be understood from the viewpoint of anabelian geometry. We formulate some new anabelian-geometric conjectures relating the tame fundamental groups of curves over algebraically closed fields of characteristic $p>0$ to the moduli spaces of curves. These conjectures are generalized versions of the weak Isom-version of the Grothendieck conjecture for curves over algebraically closed fields of characteristic $p>0$ which was formulated by Tamagawa. Moreover, we prove that the conjectures hold for certain points lying in the moduli space of curves of genus $0$.

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arXiv2022

$U(1)$-Gauge Field Theories on $G_2$-Manifolds

In this paper, we investigate two types of $U(1)$-gauge field theories on $G_2$-manifolds. One is the $U(1)$-Yang-Mills theory which admits the classical instanton solutions, we show that $G_2$-manifolds emerge from the anti-self-dual $U(1)$ instantons, which is an analogy of Yang's result for Calabi-Yau manifolds. The other one is the higher-order $U(1)$-Chern-Simons theory as a generalization of Kähler-Chern-Simons theory, by suitable choice of gauge and regularization technique, we calculate the partition function under semiclassical approximation.

math-ph↗

arXiv2021

Moduli spaces of parabolic bundles over $\mathbb{P}^1$ with five marked points

This paper considers the moduli spaces/stacks of parabolic bundles (parabolic logarithmic flat bundles and parabolic logarithmic Higgs bundles with given spectrum) of rank 2 and degree 1 over $\mathbb{P}^1$ with five marked points. The foliation and stratification structures on these moduli spaces/stacks are investigated. In particular, we confirm Simpson's conjecture for the moduli space of parabolic logarithmic flat bundles with certain non-special weight system.

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arXiv2020

Hyperbolic Superspaces and Super-Riemann Surfaces

In this paper, we will generalize some results in Manin's paper "Three-dimensional hyperbolic geometry as $\infty$-adic Arakelov geometry" to the supergeometric setting. More precisely, viewing $\mathbb{C}^{1|1}$ as the boundary of the hyperbolic superspace $\mathcal{H}^{3|2}$, we reexpress the super-Green functions on the supersphere $\hat{\mathbb{C}}^{1|1}$ and the supertorus $T^{1|1}$ by some data derived from the supergeodesics in $\mathcal{H}^{3|2}$.

math-ph↗

arXiv2020

Generalized Deligne-Hitchin Twistor Spaces: Construction and Properties

In this paper, we generalize the construction of Deligne-Hitchin twistor space by gluing two certain Hodge moduli spaces. We investigate such generalized Deligne-Hitchin twistor space as a complex analytic manifold. More precisely, we show it admits holomorphic sections with weight-one property and semi-negative energy, and it carries a balanced metric, and its holomorphic tangent bundle (for the case of rank one) is stable. Moreover, we also study the automorphism groups of the Hodge moduli spaces and the generalized Deligne-Hitchin twistor space.

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arXiv2020

On Base Change of Local Stability in Positive Characteristics

We prove that a pointed one dimensional family of varieties $\mathcal{X}\to {b\in B}$ in positive characteristics is locally stable iff the log pair $(\mathcal{X'}, \mathcal{X}'_{b'})$ arising from its base change to the perfectoid base $b'\in B_{perf}$ is log canonical.

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arXiv2018

Weak approximation for Fano complete intersections in positive characteristic

For a smooth curve $B$ over an algebraically closed field $k$, for every $B$-flat complete intersection $X_B$ in $B\times_{\text{Spec}\ k} \mathbb{P}^n_k$ of type $(d_1,\dots,d_c)$, if the Fano index is $\geq 2$ and if $\text{char}(k)>\max(d_1,\dots,d_c)$, we prove weak approximation of $\widehat{\mathcal{O}}_{B,b}$-points of $X_B$ by $k(B)$-points at all places of (strong) potentially good reduction, including all places of good reduction. The key step is the proof that such complete intersections are \emph{separably uniruled by lines}, and even \emph{separably rationally connected}, whenever smooth. We prove that the inequality is close to sharp. We prove a similar theorem for Fano manifolds of Picard number $1$ and Fano index $1$.

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