David Alfaya, Tomas L. Gomez
We prove that, given the isomorphism class of the parabolic Deligne-Hitchin moduli space over a smooth projective curve, we can recover the isomorphism class of the curve and the parabolic points.
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Let X and X' be compact Riemann surfaces of genus at least three. Let G and G' be nontrivial connected semisimple linear algebraic groups over C. If some components $M_{DH}^d(X,G)$ and $M_{DH}^{d'}(X',G')$ of the associated Deligne--Hitchin moduli spaces are biholomorphic, then X' is isomorphic to X or to the conjugate Riemann surface $\bar{X}$.
Fix integers $g\geq 3$ and $r\geq 2$, with $r\geq 3$ if $g=3$. Given a compact connected Riemann surface $X$ of genus $g$, let $\MDH(X)$ denote the corresponding $\text{SL}(r, {\mathbb C})$ Deligne--Hitchin moduli space. We prove that the complex analytic space $\MDH(X)$ determines (up to an isomorphism) the unordered pair $\{X, \overline{X}\}$, where $\overline{X}$ is the Riemann surface defined by the opposite almost complex structure on $X$.
We define the moduli problem of Hitchin pairs over Deligne-Mumford Stack and prove this moduli problem is represented by a separated and locally finitely presented algebraic space, which is considered as the moduli space of Hitchin pairs over Deligne-Mumford stack.
Generalized Deligne-Hitchin Twistor Spaces: Construction and Properties
本文通过粘合两个 Hodge 模空间构造广义 Deligne-Hitchin 扭量空间，证明其具有权一全纯截面、半负能量、平衡度量及秩一情形下全纯切丛的稳定性，并研究了相关自同构群。
Non-algebraicity of Deligne--Hitchin twistor spaces
本文证明了正亏格光滑复射影曲线上，GL(n,C) 和 SL(n,C) 的 Deligne--Hitchin 扭曲空间及其稳定轨迹（亏格≥2时）均无局部有限型复概形代数化。
Let $X$ be a compact connected Riemann surface of genus $g \geq 2$, and let ${\mathcal M}_{\rm DH}$ be the rank one Deligne-Hitchin moduli space associated to $X$. It is known that ${\mathcal M}_{\rm DH}$ is the twistor space for the hyper-Kähler structure on the moduli space of rank one holomorphic connections on $X$. We investigate the group $\operatorname{Aut}({\mathcal M}_{\rm DH})$ of all holomorphic automorphisms of ${\mathcal M}_{\rm DH}$. The connected component of $\operatorname{Aut}({\mathcal M}_{\rm DH})$ containing the identity automorphism is computed. There is a natural element o…
We study the holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface that are invariant under the natural anti-holomorphic involutions of the moduli space. Their relationships with the harmonic maps are established. As a bi-product, a question of Simpson on such sections, posed in \cite{Si2}, is answered.
It was conjectured in \cite{Namikawa_ExtendedTorelli} that the Torelli map $M_g\to A_g$ associating to a curve its jacobian extends to a regular map from the Deligne-Mumford moduli space of stable curves $\bar{M}_g$ to the (normalization of the) Igusa blowup $\bar{A}_g^{\rm cent}$. A counterexample in genus $g=9$ was found in \cite{AlexeevBrunyate}. Here, we prove that the extended map is regular for all $g\le8$, thus completely solving the problem in every genus.
Prym varieties and cubic threefolds over $\mathbb{Z}$
本文发展了特征2域上的Prym簇与三次三维簇理论，证明了光滑三次三维簇在任意域上的非有理性及Torelli定理，并解决了Deligne关于算术Torelli映射的猜想。
The twistor space of the moduli space of solutions of Hitchin's self-duality equations can be identified with the Deligne-Hitchin moduli space of $λ$-connections. We use real projective structures on Riemann surfaces to prove the existence of new components of real holomorphic sections of the Deligne-Hitchin moduli space. Applying the twistorial construction we show the existence of new hyper-Kähler manifolds associated to any compact Riemann surface of genus $g\geq2$. These hyper-Kähler manifolds can be considered as moduli spaces of (certain) singular solutions of the self-duality equations.
It has been known since the 1970s that the Torelli map $M_g \to A_g$, associating to a smooth curve its jacobian, extends to a regular map from the Deligne-Mumford compactification $\bar{M}_g$ to the 2nd Voronoi compactification $\bar{A}_g^{vor}$. We prove that the extended Torelli map to the perfect cone (1st Voronoi) compactification $\bar{A}_g^{perf}$ is also regular, and moreover $\bar{A}_g^{vor}$ and $\bar{A}_g^{perf}$ share a common Zariski open neighborhood of the image of $\bar{M}_g$. We also show that the map to the Igusa monoidal transform (central cone compactification) is NOT regul…
Mirror symmetry and Hitchin systems on DM curves: SYZ Duality
本文将Higgs丛、谱曲线与Norm映射理论推广到Deligne-Mumford曲线，并在温和条件下证明了双曲叠曲线上Higgs丛模空间的SYZ对偶。
Extending the Torelli map to alternative compactifications of the moduli space of curves
本文研究了具有轴状奇点的曲线，构造了替代紧化并证明Torelli映射可扩展，同时识别了Smyth意义下的替代紧化中轴状区域。
In "Quantization of Hitchin's Integrable System and Hecke Eigensheaves", Beilinson and Drinfeld introduced the "very good" property for a smooth complex equidimensional stack. They prove that for a semisimple complex group G, the moduli stack of G-bundles over a smooth complex projective curve X is "very good", as long as X has genus g > 1. In the case of the projective line, when g = 0, this is not the case. However, the result can sometimes be extended to the projective line by introducing additional parabolic structure at a collection of marked points and slightly modifying the definition o…
Torelli Theorem for moduli stacks of vector bundles on tame Deligne Mumford curves
本文证明了向量丛模空间的两个Torelli定理：从固定行列式模栈恢复曲线，及从扭曲模空间恢复曲线和Brauer类，方法涉及特殊线性线性化、Hitchin构造和Kummer覆盖等。
Families of Hitchin systems and N=2 theories
本研究围绕四维N=2理论关联的Hitchin系统族展开，分析其在Deligne-Mumford模空间上的全局行为，证明Hitchin基构成紧化模空间上的向量丛，还分类了幂零轨道与味对称子群的允许对。
Relative spectral correspondence for parabolic Higgs bundles and Deligne--Simpson problem
本文推广抛物Higgs丛的谱对应至相对情形，用曲面族上的线性系统刻画Hitchin像，证明OK条件是Deligne--Simpson问题可解的充分条件，并给出模空间非空性。
Let $C$ be a smooth projective curve of genus $g\geq 4$ over the complex numbers and ${\cal SU}^s_C(r,d)$ be the moduli space of stable vector bundles of rank $r$ with a fixed determinant of degree $d$. In the projectivized cotangent space at a general point $E$ of ${\cal SU}^s_C(r,d)$, there exists a distinguished hypersurface ${\cal S}_E$ consisting of cotangent vectors with singular spectral curves. In the projectivized tangent space at $E$, there exists a distinguished subvariety ${\cal C}_E$ consisting of vectors tangent to Hecke curves in ${\cal SU}^s_C(r,d)$ through $E$. Our main result…
Solutions of Hitchin's self-duality equations corresponds to special real sections in the Deligne-Hitchin moduli space -- twistor lines. A question posed by Simpson in 1997 asks whether all real sections give rise to global solutions of the self-duality equations. An affirmative answer would allow for complex analytic procedure to obtain all solutions of the self-duality equations. The purpose of this paper is to construct counter examples given by certain (branched) Willmore surfaces in $3$-space (with monodromy) via the generalized Whitham flow. Though these higher solutions do not give rise…
Let $X=\overline{X}-D$ be a smooth quasi-projective curve. In arXiv:2110.12300 we constructed a Deligne-Hitchin modui space with Hecke gauge groupoid for connections of rank $2$. We extend this construction to the case of any rank $r$, although still keeping a genericity hypothesis. The formal neighborhood of a preferred section corresponding to a tame harmonic bundle is governed by a mixed twistor structure.
In this article we give a new proof of Ngô's Geometric Stabilisation Theorem, which implies the Fundamental Lemma. This is a statement which relates the cohomology of Hitchin fibres for a quasi-split reductive group scheme $G$ to the cohomology of Hitchin fibres for the endoscopy groups $H_κ$. Our proof avoids the Decomposition and Support Theorem, instead the argument is based on results for $p$-adic integration on coarse moduli spaces of Deligne-Mumford stacks. Along the way we establish a description of the inertia stack of the (anisotropic) moduli stack of $G$-Higgs bundles in terms of end…
Semiclassical Limits of Strongly Parabolic Higgs Bundles and Hyperpolygon Spaces
本文研究强抛物Higgs丛的半经典极限，揭示超多边形空间的几何结构及其与Hitchin度量的联系。
A transfer principle: from periods to isoperiodic foliations
本研究分类了Hodge丛上等周期叶状结构的叶闭包并证明其遍历性，借助模空间Torelli覆叠上周期映射纤维的连通性推导结果，还得到椭圆曲线原始分支覆叠Hurwitz空间的拓扑推论，同时发展了增广Torelli空间理论。
A ball quotient parametrizing trigonal genus 4 curves
本文证明带 $g^1_3$ 的亏格4曲线模空间是9维Deligne-Mostow球商的 $\ rac{1}{2}(3^{10}-1)$ 次覆盖，并给出不依赖K3曲面和Torelli定理的初等构造。
Let $X$ be a quasi-projective curve, compactified to $(Y,D)$ with $X=Y-D$. We construct a Deligne-Hitchin twistor space out of moduli spaces of framed $λ$-connections of rank $2$ over $Y$ with logarithmic singularities and quasi-parabolic structure along $D$. To do this, one should divide by a Hecke-gauge groupoid. Tame harmonic bundles on $X$ give preferred sections, and the relative tangent bundle along a preferred section has a mixed twistor structure with weights $0,1,2$. The weight $2$ piece corresponds to the deformations of the KMS structure including parabolic weights and the residues …
This work is the geometric part of our proof of the weighted fundamental lemma, which is an extension of Ngô Bao Châu's proof of the Langlands-Shelstad fundamental lemma. Ngô's approach is based on a study of the elliptic part of the Hichin fibration. The total space of this fibration is the algebraic stack of Hitchin bundles and its base space is the affine space of "characteristic polynomials". Over the elliptic set, the Hitchin fibration is proper and the number of points of its fibers over a finite field can be expressed in terms of orbital integrals. In this paper, we study the Hitchin fi…
Number of cuspidal automorphic representations and Hitchin's moduli spaces
本文研究有限域上曲线的函数域中、在指定点集具有固定局部Deligne-Lusztig诱导表示的尖点自守表示计数问题，通过Hitchin模叠的群胚基数给出表达式，并利用Arthur-Selberg迹公式变体证明了几何侧的消失性结果。
The Dirac-Higgs complex and categorification of (BBB)-branes
该研究遵循Gaiotto的建议，引入德利涅叠构建(BBB)-膜的范畴化框架，构造了一类(BBB)-膜并探讨其与几何朗兰兹对应中威尔逊函子的关联。
Generalized Calabi-Yau structures, a notion recently introduced by Hitchin, are studied in the case of K3 surfaces. We show how they are related to the classical theory of K3 surfaces and to moduli spaces of certain SCFT as studied by Aspinwall and Morrison. It turns out that K3 surfaces and symplectic structures are both special cases of this general notion. The moduli space of generalized Calabi-Yau structures admits a canonical symplectic form with respect to which the moduli space of symplectic structures is Lagrangian. The standard theory of K3 surfaces implies surjectivity of the period …
The Hodge filtration on nonabelian cohomology
这部分是一篇关于nonabelian霍奇理论的调查文章，但我们也证明了只在其他地方公布的结果。在引言中，我们讨论了有关该主题的广泛最新工作并提供了一些参考。在本文的主体部分，我们讨论了Corlette的非交换Hodge定理，Hitchin的四元数结构上的模空间表示的$π_1(X)$（对于一个紧凑的Kähler流形$X$），和Deligne的建设所产生的扭量空间。然后，我们提到的解释德利涅的空间$λ$ -联络的霍奇过滤nonabelian德拉姆上同调$M_{DR}(X,G)= H^1(X,G)$。我们继续证明这个霍奇过滤的几个属性，如格里菲斯的横截性和规律性的高斯-马宁连接;一个本地平凡属性来自高盛和米尔森的奇点分析;和一个重量属性来自朗顿的理论。利用Hodge滤子的权性质构造了$M_{DR}$的紧化.最后，我们定义了Noether-Lefschetz轨迹的一个非交换形式，并证明了如果纤维化的基是紧的，则它是代数的。我们提出的开放性问题研究退化的nonabelian霍奇结构足够好，能够证明代数的诺特-莱夫谢茨轨迹，即使当基地是quasiprojective。
The notion of flat $λ$-connections as the interpolation of usual flat connections and Higgs fields was suggested by Deligne and further studied by Simpson. Mochizuki established the Kobayashi--Hitchin-type theorem for $λ$-flat bundles ($λ\neq 0$), which is called the Mochizuki correspondence. In this paper, on the one hand, we generalize Mochizuki's result to the case when the base being a compact balanced manifold, more precisely, we prove the existence of harmonic metrics on stable $λ$-flat bundles ($λ\neq 0$). On the other hand, we study two applications of the Simpson--Mochizuki correspond…
Deformation theory of parabolic representation pairs
本文研究了抛物表示对的变形理论，通过引入相关变种和群oids，证明了DGLA的混合形式，并建立了多项稳定的抛物表示对的Kobayashi-Hitchin型定理。
Purity and 2-Calabi-Yau categories
本文针对具有良好模空间的2-Calabi-Yau范畴对象栈，利用形式性和étale邻域定理建立混合Hodge模复形及Borel-Moore同调的纯度，进而证明分解定理、定义尖点上同调，并推广到无良好模空间的凝聚层栈及奇异Higgs层栈。
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