arXiv subjects

Tianzhi Hu

Publications and source records attributed to Tianzhi Hu.

8 recordsLinked to original sources

arXiv2026

A numerically flat rank-two bundle without a holomorphic connection on a $\partial\bar\partial$-threefold

We construct a numerically flat holomorphic vector bundle of rank two on a compact complex threefold satisfying the ordinary $\partial\bar\partial$-lemma and prove that it admits no holomorphic connection. This gives, in particular, a negative answer to a question posed by Cao--Deng--Matsumura. In contrast, for compact simply connected complex manifolds $X$, we prove that the answer is affirmative if $H^1(X,\mathcal O_X)=0$, and hence if the Frölicher spectral sequence of $X$ degenerates at $E_1$.

math.CV↗

arXiv2026

Sharp degree bound for rational proper maps from $\mathbb B^2$ to $\mathbb B^4$

We prove D'Angelo's degree conjecture for rational proper holomorphic maps from $\mathbb{B}^2$ to $\mathbb{B}^4$, establishing the sharp degree bound of five. Suppose, to the contrary, that a rational proper map of degree six exists. We associate to the map a characteristic number measuring the degeneracy of its projective differential data. A global intersection-theoretic computation determines this number exactly, while a local analysis along the degeneracy locus yields a strictly larger lower bound for the same quantity. This contradiction excludes degree six and proves the conjectured bound.

math.CV↗

arXiv2026

Holomorphic isomonodromic deformations of Higgs bundles: absolute lifting, spectral flatness, and nilpotent rigidity

Let $f:X\to S$ be a smooth projective family, and fix a semisimple flat bundle on a fiber $X_0$. Its isomonodromic deformation determines a holomorphic section $σ_{\mathrm{dR}}:S\to M_{\mathrm{dR}}(X/S)$ of the relative de Rham moduli space. Applying the relative non-abelian Hodge correspondence fiberwise gives a section $σ_{\mathrm{Dol}}:S\to M_{\mathrm{Dol}}(X/S)$, whose value at each $s\in S$ is the Higgs bundle corresponding to the flat bundle $σ_{\mathrm{dR}}(s)$. Unlike $σ_{\mathrm{dR}}$, the section $σ_{\mathrm{Dol}}$ is in general only real analytic. We study the geometric consequences of its holomorphicity along a complex analytic subvariety. First, we prove an absolute lifting theorem: the relative isomonodromic Higgs bundle admits an absolute Higgs lift, and the fiberwise harmonic metrics can be modified to solve the Hitchin-Simpson equation on the total space. Second, we prove that the spectral one-form on every resolved irreducible component of the relative spectral scheme is Gauss-Manin flat. As an application, we prove the nilpotent rigidity conjecture of Hu-Sun-Yang-Zuo: if the initial Higgs field is nilpotent, then the Higgs field remains nilpotent along the entire holomorphic isomonodromic locus.

math.AG↗

arXiv2026

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.

math.AG↗

arXiv2026

Non-Abelian Kodaira-Spencer Map and non-existence of holomorphic isomonodromic deformation of Higgs bundles over Teichmüller 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 Teichmüller 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 Teichmüller spaces.

math.AG↗

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.

math.AG↗

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 $σ_{Dol}:S\to M_{Dol}(X/S)$ 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 $σ_{Dol}$ 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 $σ_{Dol}$ 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.

math.AG↗

arXiv2026

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 André-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 $\mathbb{P}^4$.

math.AG↗