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Kang Zuo

Publications and source records attributed to Kang Zuo.

At least 19 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

Rigidity Criterion for Certain Calabi-Yau Families

We prove a new rigidity criterion for families of polarized Calabi--Yau manifolds. Motivated by known non-rigid examples, we conjecture that a family over a quasi-projective curve is rigid if, near a boundary point, the total space is smooth, the relative canonical bundle is trivial, and the boundary fiber contains an isolated singular point. We verify this conjecture when one such isolated singularity has a concentrated mixed Hodge spectrum, a class including ordinary double points and cusps. The proof combines a local vanishing-cycle analysis with a global tensor-product decomposition of the associated variation of Hodge structures.

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arXiv2025

Non-Abelian Kodaira-Spencer Map and non-existence of holomorphic isomonodromic deformation of Higgs bundles over Teichm\"uller spaces

We define the isomonodromic deformation of a Higgs bundle on a compact Riemann surface via the Hitchin--Simpson correspondence and the isomonodromic deformation of the associated local system. This construction yields a real-analytic section of the relative Dolbeault moduli space and hence a real-analytic foliation, generalizing the Betti foliation arising from the Betti map in the study of abelian schemes. We give cohomological expressions for the holomorphic and anti-holomorphic derivatives of the isomonodromic deformation and use the latter to extend the classical non-abelian Kodaira--Spencer map. We prove that if the isomonodromic deformation of a graded Higgs bundle is non-holomorphic, then the deformed Higgs field is non-nilpotent. We also give a short new proof of the non-existence of holomorphic isomonodromic deformations for generic Higgs bundles over Teichm\"uller space $\mathcal T_g$, previously established in \cite{biswas}. This shows that global holomorphicity imposes strong restrictions on the initial Higgs bundle. Motivated by this observation, we prove, under suitable numerical conditions, that non-nilpotent or non-unitary Higgs bundles have non-holomorphic isomonodromic deformations over $\mathcal T_g$. These results may be viewed as analogues of the Landesman--Litt finite-image theorem for MCG-finite representations \cite{LL}. This paper synthesizes and refines our two earlier preprints \cite{HSZ,HSZII} (arXiv:2511.14272 and arXiv:2512.15478), and makes further progress on the non-existence problem for holomorphic isomonodromic deformations of higher rank Higgs bundles over Teichm\"uller spaces.

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arXiv2025

Kodaira-Spencer Map on the Hitchin-Simpson Correspondence

We define the isomonodromic deformation of a Higgs bundle over a compact Riemann surface via the Hitchin-Simpson correspondence and the isomonodromic deformation of a local system. This deformation defines a real analytic section of the relative Dolbeault moduli space, yielding a real analytic foliation on this moduli. This foliation generalizes the Betti foliation defined by the Betti map in the study of abelian schemes. We provide a precise form for the holomorphic and anti-holomorphic derivatives of the isomonodromic deformation of a Higgs bundle. Subsequently, we extend the classical non-abelian Kodaira-Spencer map using the anti-holomorphic derivative. Additionally, we prove that if the isomonodromic deformation of a graded Higgs bundle is not holomorphic, then the isomonodromically deformed Higgs field is non-nilpotent.

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arXiv2025

Logarithmic Crystalline Representations

In 1989, Faltings proved the comparison theorem between \'etale 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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arXiv2024

On the distribution of non-rigid families in the moduli spaces

This paper investigates the distribution of non-rigid families in a moduli space $\mathcal{M}$ of polarized projective manifolds for which the infinitesimal Torelli theorem holds. Guided by the analogy with unlikely intersection in Shimura varieties, we show that the image of any non-rigid classifying morphisms into $\mathcal{M}$ is contained in the Hodge locus as long as the derived Mumford-Tate group is $\mathbb{Q}$-simple and the period map is generically finite. If moreover the period domain is not Hermitian of rank at least 2, then the Hodge locus can be replaced by a closed subscheme, which yields a finiteness theorem of geometric Bombieri-Lang type. Inspired by the Zilber-Pink conjecture, we also characterize the geometry of non-rigid locus by the specialness of bi-Hom schemes and the finiteness of "structurally-atypical" intersections. Finally, we specialize to the moduli spaces of polarized Calabi-Yau manifolds, formulate an unobstructedness conjecture for non-rigid maps which implies the specialness of bi-Hom schemes, prove a geometric Andr\'e-Oort theorem describing the Zariski closure of non-rigid locus, and test the theory and the conjecture for the explicit Viehweg-Zuo family of Calabi--Yau quintics in $\mathbb{P}^4$.

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arXiv2023

Finiteness of pointed maps to moduli spaces of polarized varieties

We establish a finiteness result for pointed maps to the base space $U$ of a smooth projective family of varieties with maximal variation in moduli. For its proof, we establish the rigidity of pointed maps to a (not necessarily compact) variety which is hyperbolic modulo a proper closed subset. Together with Viehweg's hyperbolicity conjecture on the bigness of log-canonical bundles of moduli spaces, resolved by Campana-Paun, we derive an optimal dimension bound on the Hom scheme from a curve to $U$ among other applications.

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arXiv2023

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 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

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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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

Higgs Bundles in Geometry and Arithmetic

We introduce the notions of deformation Higgs bundle and Riemann-Finsler metric on the moduli space of polarized varieties. We also use the Higgs-de Rham flow in the p-adic setting. These are the key novelties in our program.

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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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arXiv2021

Picard theorems for moduli spaces of polarized varieties

As a result of our study of the hyperbolicity of the moduli space of polarized manifold, we give a general big Picard theorem for a holomorphic curve on a log-smooth pair $(X,D)$ such that $W=X\setminus D$ admits a Finsler pseudometric that is strongly negatively curved when pulled back to the curve. We show, by some refinements of the classical Viehweg-Zuo construction, that this latter condition holds for the base space $W$, if nonsingular, of any algebraic family of polarized complex projective manifolds with semi-ample canonical bundles whose induced moduli map $ϕ$ to the moduli space of such manifolds is generically finite and any $ϕ$-horizontal holomorphic curve in $W$. This yields the big Picard theorem for any holomorphic curves in the base space $U$ of such an algebraic family by allowing this base space to be singular but with generically finite moduli map. An immediate and useful corollary is that any holomorphic map from an algebraic variety to such a base space $U$ must be algebraic, i.e., the corresponding holomorphic family must be algebraic. We also show the related algebraic hyperbolicity property of such a base space $U$, which generalizes previous Arakelov inequalities and weak boundedness results for moduli stacks and offers, in addition to the Picard theorem above, another evidence in favor of the hyperbolic embeddability of such an $U$.

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