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Mao Sheng

Publications and source records attributed to Mao Sheng.

At least 19 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

Uniformization as Tannakian Reconstruction

Classical hyperbolic uniformization identifies every hyperbolic log-orbi curve with a compactified quotient of the upper half-plane by a cofinite Fuchsian lattice. The lattice is unique up to conjugacy. We reconstruct it intrinsically. For each hyperbolic log-orbi curve C we construct a canonical maximal principal PSL2-Higgs object. Etale-locally it comes from the standard square-root SL2-model. The central mu2 ambiguity disappears after passage to PSL2. Using vector tame non-abelian Hodge theory and regular-singular Riemann--Hilbert as input we assemble the required principal realizations Tannakianly. Parahoric structures encode the orbifold and cusp data on the coarse curve. After choosing a base point and conjugating the Betti realization is represented by a discrete faithful finite-covolume representation whose image is the uniformizing lattice. Compatibility with finite etale pullback makes the lattice construction a quasi-inverse to the compactified quotient functor. Thus classical uniformization is recast as an intrinsic Tannakian reconstruction theorem. We also identify finite etale covers with finite continuous sets for the profinite completion of the reconstructed lattice. After fixing a separable closure and the resulting geometric generic point we recover the absolute Galois group of the function field of C as the inverse limit of the based etale fundamental groups of orbifold models over C.

math.AG↗
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

Nonabelian Kodaira-Spencer maps

We give an explicit formula of the associated graded map to the nonabelian Gauss-Manin connection with respect to the nonabelian Hodge filtration.

math.AG↗
arXiv2025

Strong semistability of Higgs bundles over curves

In this paper we complete the study of the Lan-Sheng-Zuo conjecture proposed in arXiv:1210.8280 for the curve case. Precisely, we prove that every semistable Higgs bundle is strongly semistable for curves of genus $g\leq 1$, and over any curves of genus $g\ge2$ construct explicit examples of semistable Higgs bundles of arbitrary big rank (the first example is $p=2,r=3$) which are not strongly semistable. These results are complementary to the strongly semistability theorem of Lan-Sheng-Yang-Zuo and Langer for semistable Higgs bundles of small rank.

math.AG↗
arXiv2024

The small $p$-adic Simpson correspondence in the semi-stable reduction case

We generalize several known results on small Simpson correspondence for smooth formal schemes over $\calO_C$ to the case for semi-stable formal schemes. More precisely, for a liftable semi-stable formal scheme $\frakX$ over $\calO_C$ with generic fiber $X$, we establish (1) an equivalence between the category of Hitchin-small integral $v$-bundles on $X_{v}$ and the category of Hitchin-small Higgs bundles on $\frakX_{\et}$, generalizing the previous work of Min--Wang, and (2) an equivalence between the moduli stack of $v$-bundles on $X_{v}$ and the moduli stack of rational Higgs bundles on $\frakX_{\et}$ (equivalently, moduli stack of Higgs bundles on $X_{\et}$), generalizing the previous work of Ansch\"utz--Heuer--Le Bras.

math.AG↗
arXiv2024

A Nonabelian Hodge Correspondence for Principal Bundles in Positive Characteristic

In this paper, we prove a nonabelian Hodge correspondence for principal bundles on a smooth variety $X$ in positive characteristic, which generalizes the Ogus-Vologodsky correspondence for vector bundles. Then we extend the correspondence to logahoric torsors over a log pair $(X,D)$, where $D$ a reduced normal crossing divisor in $X$. As an intermediate step, we prove a correspondence between principal bundles on root stacks $\mathscr{X}$ and parahoric torsors on $(X,D)$, which generalizes the correspondence on curves given by Balaji--Seshadri to higher dimensional case.

math.AG↗
arXiv2024

On the Existence of Gr-semistable Filtrations of Orthogonal/Symplectic $λ$-connections

In this paper, we study the existence of gr-semistable filtrations of orthogonal/symplectic $λ$-connections. It is known that gr-semistable filtrations always exist for flat bundles in arbitrary characteristic. However, we found a counterexample of orthogonal flat bundles of rank 5 in positive characteristic. The central new idea in this example is the notion of quasi gr-semistability for orthogonal/symplectic $λ$-connections. We establish the equivalence between gr-semistability and quasi gr-semistablity for an orthogonal/symplectic $λ$-connection. This provides a way to determine whether an orthogonal/symplectic $λ$-connection is gr-semistable. As an application, we obtain a characterization of gr-semistable orthogonal $λ$-connections of rank $\leq 6$.

math.AG↗
arXiv2023

On the hydraulic fracturing in naturally-layered porous media using the phase field method

In the hydraulic fracturing of natural rocks, understanding and predicting crack penetrations into the neighboring layers is crucial and relevant in terms of cost-efficiency in engineering and environmental protection. This study constitutes a phase field framework to examine hydraulic fracture propagation in naturally-layered porous media. Biot's poroelasticity theory is used to couple the displacement and flow field, while a phase field method helps characterize fracture growth behavior. Additional fracture criteria are not required and fracture propagation is governed by the equation of phase field evolution. Thus, penetration criteria are not required when hydraulic fractures reach the material interfaces. The phase field method is implemented within a staggered scheme that sequentially solves the displacement, phase field, and fluid pressure. We consider the soft-to-stiff and the stiff-to-soft configurations, where the layer interface exhibits different inclination angles $θ$. Penetration, singly-deflected, and doubly-deflected fracture scenarios can be predicted by our simulations. In the soft-to-stiff configuration, $θ=0^\circ$ exhibits penetration or symmetrical doubly-deflected scenarios, and $θ=15^\circ$ exhibits singly-deflected or asymmetric doubly-deflected scenarios. Only the singly-deflected scenario is obtained for $θ=30^\circ$. In the stiff-to-soft configuration, only the penetration scenario is obtained with widening fractures when hydraulic fractures penetrate into the soft layer.

physics.geo-ph↗
arXiv2023

A torsion property of the zero of Kodaira-Spencer over $\mathbb{P}^1$ removing four points

We establish a torsion theorem to the effect that the unique zero of the Kodaira-Spencer map attached to a certain quasi-semistable family of complex projective varieties over the complex projective line is the image of a torsion point of an elliptic curve under the natural projection. The proof is a mod $p$ argument and requires a density one set of primes. There are three essential ingredients in the proof: a solution to the conjecture of Sun-Yang-Zuo, which constitutes the principal part of the paper, Pink's theorem, and Higgs periodicity theorem.

math.AG↗
arXiv2023

Intersection de Rham complexes in positive characteristic

We establish a positive characteristic analogue of intersection cohomology theory for variations of Hodge structure. It includes: a) the de Rham-Higgs comparison theorem for the intersection de Rham complex; b) the $E_1$-degeneration theorem for the intersection de Rham complex of a periodic de Rham bundle; c) the Kodaira-Saito vanishing theorem for the intersection cohomology groups of a periodic Higgs bundle. These results generalize the decomposition theorem of Deligne-Illusie and the de Rham-Higgs theorem of Ogus-Vologodsky, the $E_1$-degneration theorem of Deligne-Illusie, Illusie, Faltings and the Kodaira-Saito vanishing theorem of Arapura. As an application, we give an algebraic proof of the $E_1$-degeneration theorem due to Cattani-Kaplan-Schmid and Kashiwara-Kawai, and the vanishing theorem of Saito for VHSs of geometric origin.

math.AG↗
arXiv2023

An explicit infinite homotopy in nonabelian Hodge theory in positive characteristic

This short note is extracted from Section 3 and Appendix of the paper entitled with Intersection de Rham complexes in positive characteristic by the same named authors, where an explicit infinite homotopy from a Higgs complex to the Frobenius pushforward of the corresponding de Rham complex in positive characteristic has been provided. The verification details, which are omitted therein, are provided here.

math.AG↗
arXiv2022

Periodic de Rham bundles over curves

In this article, we introduce the notion of periodic de Rham bundles over smooth complex curves. We prove that motivic de Rham bundles over smooth complex curves are periodic. We conjecture that irreducible periodic de Rham bundles over smooth complex curves are motivic. We show that the conjecture holds for rank one objects and rigid objects.

math.AG↗
arXiv2022

Periodicity of Hitchin's uniformizing Higgs bundles

We link the periodicity of Hitchin's uniformizing Higgs bundle with the arithmetic geometry of its underlying curve. Some new relations are discovered. We also speculate on the whole class of periodic Higgs bundles.

math.AG↗
arXiv2021

Output fusion of MPC and PID and its application in intelligent layered water injection of oilfield

To improve the dynamic response performance of wave code communication in intelligent layered water injection of oilfield, this paper proposes an output optimal fusion control method based on MPC-PID. Firstly, depending on the well structure and the flow-pressure characteristics of the layer, the steady-state model between the differential pressure and flow of the whole well and different layer sections is established for layered water injection, and the corresponding wave code amplitude at the steady-state operating point of different layer sections is solved, the numerical calculation verifies that the increase of the nozzle opening in a single layer section will drive the pressure and flow curve of the whole well downward. Secondly, combining the dynamic response characteristics and steady-state model of the whole-well water distribution equipment, a dynamic model of layered intelligent water injection is established, and the generation process of the wave code is defined; Finally, the MPC-PID optimal fusion control algorithm structure is designed to derive the fusion control law that minimizes the cost function under fixed weights, , and the optimal weights are calculated by combining the internal model structure of controller, so the optimization performance of each algorithm in the optimal fusion control is balanced. By analyzing the control simulation results, the fast response characteristics of the fusion control method are verified. Meanwhile, the simulation comparison experiments of fast wave code communication under different methods are conducted with the actual working conditions, the results show that the fusion control method has both fast tracking control capability and strong robustness, which effectively enhances the efficiency of wave code communication and shortens the wave code operation time.

eess.SY↗