arXiv subjects

Jinbang Yang

Publications and source records attributed to Jinbang Yang.

16 recordsLinked to original sources

arXiv2026

Isomonodromic deformations of Higgs bundles and characterization of the non-abelian Noether--Lefschetz locus

Let $f:X\to S$ be a smooth proper family of smooth projective varieties. An irreducible complex local system on a fiber admits an isomonodromic deformation, hence determines a holomorphic section of the relative de Rham moduli space. Applying the relative non-abelian Hodge correspondence produces a real-analytic section $\sigma_{Dol}:S\to M_{Dol}(X/S)$ of the relative Dolbeault moduli space. In this paper, we investigate when this real-analytic section is holomorphic. The first approach uses the first-order infinitesimal deformation: we prove a Cauchy--Riemann type criterion showing that holomorphicity in a tangent direction of $S$ is measured by the composition of the Kodaira--Spencer map with the non-abelian Higgs field. The second approach involves higher-order derivatives: after restricting $\sigma_{Dol}$ to infinitesimal thickenings of the reference point in $S$, we introduce obstruction classes measuring the failure of holomorphicity and relate them to the Taylor expansion of the harmonic metric. We apply these criteria to three problems. First, we study the interaction between the $\mathbb C^*$-action on Higgs bundles and isomonodromic deformations.Second, for an initial polarized complex variation of Hodge structures, we consider the associated non-abelian Noether--Lefschetz locus. We prove that this locus is precisely the maximal complex analytic subvariety of $S$ on which the real-analytic isomonodromic deformation $\sigma_{Dol}$ becomes holomorphic. Both the first-order and higher-order methods yield proofs of this characterization. Lastly, we prove that if the initial Higgs bundle is generically regular nilpotent and the isomonodromic deformation is holomorphic, then every member of the family is represented by a nilpotent Higgs bundle.

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arXiv2026

Higher order isomonodromic deformation of Higgs bundles and a characterization of the non-abelian Noether-Lefschetz locus

The purpose of this paper is to establish a local theory of the non-abelian Noether--Lefschetz locus. Given a family of projective manifolds over a complex variety $S$, the isomonodromic deformation of the initial $\mathbb C$-PVHS defines a holomorphic family of flat bundles and defines a real analytic family of Higgs bundles by the non-abelian Hodge correspondence. The non-abelian Noether--Lefschetz locus exactly consists of those points in $S$ on which the isomonodromic deformed Higgs bundle underlies a graded structure. Esnault-Kerz ask whether the non-abelian Noether--Lefschetz locus is precisely the maximal complex analytic subvariety on which the real analytic isomonodromic deformation of Higgs bundles becomes holomorphic. Our main result gives an affirmative answer to this question. The proof is based on the deformation equation of the harmonic metric solved by the non-abelian Hodge correspondence, and we use it to study higher order deformation class of the isomonodromic deformation of a graded Higgs bundle, which is expressed in terms of the differential graded Lie algebra of the joint real analytic deformation. We introduce a sequence of obstruction classes measuring the failure of holomorphicity and show that their vanishing forces the graded structure to lift to arbitrary finite order. This yields a local characterization of the non-abelian Noether--Lefschetz locus in terms of the holomorphicity of the isomonodromic deformation of Higgs bundles.

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arXiv2026

Logarithmic Crystalline Representations

In 1989, Faltings proved the comparison theorem between étale cohomology and crystalline cohomology by studying Fontaine-Faltings modules and crystalline representations. In his paper, he mentioned these modules and representations can be extended to the logarithmic context, but without detail. This note aims to explicitly present the construction of logarithmic Fontaine-Faltings modules and logarithmic crystalline representations.

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arXiv2025

A Lefschetz theorem for crystalline representations

As a corollary of nonabelian Hodge theory, Simpson proved a strong Lefschetz theorem for complex polarized variations of Hodge structure. We show an arithmetic analog. Our primary technique is $p$-adic nonabelian Hodge theory. Conditional on certain foundational results in \emph{logarithmic} $p$-adic Hodge theory, we also show a logarithmic analog.

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arXiv2025

Parabolic Crystalline Representations

The theory of crystalline representations was established by Fontaine and Laffaille, Faltings, and others. In this paper, we develop a parabolic version of this theory. The key point is the construction of the parabolic version of Fontaine-Faltings modules and Faltings' $\mathbb D$-functor. The theory of Higgs-de Rham flows can be used to efficiently construct crystalline representations. We have established a parabolic version and utilized it to construct infinitely many crystalline representations. The twisted versions discussed in Sun, Yang, and Zuo's work can be seen as a special case, where the parabolic weights are equal at every infinity point.

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arXiv2023

Constructing abelian varieties from rank 2 Galois representations

Let $U$ be a smooth affine curve over a number field $K$ with a compactification $X$ and let $\mathbb L$ be a rank $2$, geometrically irreducible $\bar{\mathbb Q}_\ell$-local system on $U$ with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field $E\subset \bar{\mathbb Q}_\ell$, and has bad, infinite reduction at some closed point $x$ of $X\setminus U$. We show that $\mathbb L$ occurs as a summand of the cohomology of a family of abelian varieties over $U$. The argument follows the structure of the proof of a recent theorem of Snowden-Tsimerman, who show that when $E=\mathbb Q$, then $\mathbb L$ is isomorphic to the cohomology of an elliptic curve $E_U\rightarrow U$.

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arXiv2023

Constructing families of abelian varieties of $\text{GL}_2$-type over $4$-punctured complex projective line via $p$-adic Hodge theory and Langlands correspondence and application to algebraic solutions of Painleve VI equation

we construct infinitely many non-isotrivial families of abelian varieties of $GL_2$-type over four punctured projective lines with bad reduction of type-$(1/2)_\infty$ via $p$-adic Hodge theory and Langlands correspondence. They lead to algebraic solutions of Painleve VI equation. Recently Lin-Sheng-Wang proved the conjecture on the torsionness of zeros of Kodaira-Spencer maps of those type families. Based on their theorem we show the set of those type families of abelian varieties is exactly parameterized by torsion sections of the universal family of elliptic curves modulo the involution. After our paper submitted in arxiv, Lam-Litt gave a totally new construction of those abelian schemes by applying Katz's middle convolution.

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arXiv2023

Constructing algebraic solutions of Painleve VI equation from $p$-adic Hodge theory and Langlands Correspondence

We construct infinitely many non-isotrivial families of abelian varieties over given four punctured projective lines. These families lead to algebraic solutions of Painleve VI equation. Finally, based on a recent paper by Lin-Sheng-Wang, we prove a complete characterization for the locus of motivic Higgs bundles in the moduli space as fixed points of an ``additive'' self-map. This is a note based on the lecture given by the second named author on 04 Nov. 2022 at Tsinghua University.

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arXiv2022

Strict Arakelov inequality for a family of varieties of general type

Let $f:\, X\to Y$ be a semistable non-isotrivial family of $n$-folds over a smooth projective curve with discriminant locus $S \subseteq Y$ and with general fibre $F$ of general type. We show the strict Arakelov inequality \[\frac{\mathrm{deg}\, f_*ω_{X/Y}^ν}{\mathrm{rank}\, f_*ω_{X/Y}^ν} < {nν\over 2}\cdot\mathrm{deg}\,Ω^1_Y(\log S),\] for all $ν\in \mathbb N$ such that the $ν$-th pluricanonical linear system $|ω^ν_F|$ is birational. This answers a question asked by Möller, Viehweg and the third named author.

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arXiv2021

A note on p-adic Simpson correspondence

Given a proper smooth $p$-adic variety, we show a comparison theorem for the $p$-adic Simpson correspondence constructed by Faltings and Riemann-Hilbert correspondence constructed by Scholze. As an application we formulate a sufficient condition for $\overline{\mathbb Q}_p$-local system being de Rham. We study a $p$-adic analogue of Simpson's $\mathbb C^*$-action on the set of isomorphism classes of Higgs bundles and the corresponding Galois action on the set of isomorphism classes of generalized representations of the étale fundamental group.

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arXiv2020

Finiteness of logarithmic crystalline representations II

Let $K$ be an unramified $p$-adic local field and let $W$ be the ring of integers of $K$. Let $(X,S)/W$ be a smooth proper scheme together with a simple normal crossings divisor and fix positive integers $r$ and $f$. We show that the set of absolutely irreducible representations $π_1(X_{\bar K})\rightarrow \mathrm{GL}_r(\mathbb{Z}_{p^f})$ that come from log crystalline $\mathbb Z_{p^f}$-local systems over $(X_K,S_K)$ of rank $r$ is finite. The proof uses $p$-adic nonabelian Hodge theory and a finiteness result due Abe/Lafforgue.

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arXiv2020

Finiteness of logarithmic crystalline representations

Let $K$ be an unramified $p$-adic local field and let $W$ be the ring of integers of $K$. Let $(X,S)/W$ be a smooth proper scheme together with a normal crossings divisor. We show that there are only finitely many log crystalline $\mathbb Z_{p^f}$-local systems over $X_K\setminus S_K$ of given rank and with geometrically absolutely irreducible residual representation, up to twisting by a character. The proof uses $p$-adic nonabelian Hodge theory and a finiteness result due Abe/Lafforgue.

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arXiv2020

Deformation theory of periodic Higgs-de Rham flows

In this note we study the deformation theory of periodic (logarithmic) Higgs-de Rham flows. Under suitable numerical assumptions, this is equivalent to the deformation theory of torsion (logarithmic) Fontaine-Faltings modules. As an application, we formulate an \emph{ordinarity} condition, which provides a sufficient condition for a $p^n$-torsion crystalline representation to deform to a $p^{n+1}$-torsion crystalline representation.

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arXiv2019

Projective Crystalline Representations of Étale Fundamental Groups and Twisted Periodic Higgs-de Rham Flow

This paper contains three new results. {\bf 1}.We introduce new notions of projective crystalline representations and twisted periodic Higgs-de Rham flows. These new notions generalize crystalline representations of étale fundamental groups introduced in [7,10] and periodic Higgs-de Rham flows introduced in [19]. We establish an equivalence between the categories of projective crystalline representations and twisted periodic Higgs-de Rham flows via the category of twisted Fontaine-Faltings module which is also introduced in this paper. {\bf 2.}We study the base change of these objects over very ramified valuation rings and show that a stable periodic Higgs bundle gives rise to a geometrically absolutely irreducible crystalline representation. {\bf 3.} We investigate the dynamic of self-maps induced by the Higgs-de Rham flow on the moduli spaces of rank-2 stable Higgs bundles of degree 1 on $\mathbb{P}^1$ with logarithmic structure on marked points $D:=\{x_1,\,...,x_n\}$ for $n\geq 4$ and construct infinitely many geometrically absolutely irreducible $\mathrm{PGL_2}(\mathbb Z_p^{\mathrm{ur}})$-crystalline representations of $π_1^\text{et}(\mathbb{P}^1_{\mathbb{Q}_p^\text{ur}}\setminus D)$. We find an explicit formula of the self-map for the case $\{0,\,1,\,\infty,\,λ\}$ and conjecture that a Higgs bundle is periodic if and only if the zero of the Higgs field is the image of a torsion point in the associated elliptic curve $\mathcal{C}_λ$ defined by $ y^2=x(x-1)(x-λ)$ with the order coprime to $p$.

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arXiv2017

Base change of twisted Fontaine-Faltings modules and Twisted Higgs-de Rham flows over very ramified valuation rings

In this short notes, we prove a stronger version of Theorem 0.6 in our previous paper arXiv:1709.01485: Given a smooth log scheme $(\mathcal{X} \supset \mathcal{D})_{W(\mathbb{F}_q)}$, each stable twisted $f$-periodic logarithmic Higgs bundle $(E,θ)$ over the closed fiber $(X \supset D)_{\mathbb{F}_q}$ will correspond to a $\mathrm{PGL}_r(\mathbb{F}_{p^f})$-crystalline representation of $π_1((\mathcal{X} \setminus \mathcal{D})_{W(\mathbb{F}_q)[\frac{1}{p}]})$ such that its restriction to the geometric fundamental group is absolutely irreducible.

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arXiv2015

On a conjecture of Wan about limiting Newton polygons

We show that for a monic polynomial $f(x)$ over a number field $K$ containing a global permutation polynomial of degree $>1$ as its composition factor, the Newton Polygon of $f\mod\mathfrak p$ does not converge for $\mathfrak p$ passing through all finite places of $K$. In the rational number field case, our result is the "only if" part of a conjecture of Wan about limiting Newton polygons.

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