Michael Lee Baker
We define the universal Hitchin discriminants in the case of $\mathrm{Sp}(2n)$ Hitchin spectral covers, describe their components, and express their divisor classes in terms of standard generators of the rational Picard group of the moduli spaces of such spectral covers with variable base curve.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We define two tau functions, $τ$ and $\hatτ$ , on moduli spaces of spectral covers of $GL(n)$ Hitchin's systems. Analyzing the properties of $τ$, we express the divisor class of the universal Hitchin's discriminant in terms of standard generators of the rational Picard group of the moduli spaces of spectral covers with variable base. The function $\hatτ$ is used to compute the divisor of canonical 1-forms with multiple zeros.
We continue the study of the rational Picard group of the moduli space of Hitchin's spectral covers started in P. Zograf's and D. Korotkin's work [11]. In the first part of the paper we expand the ``boundary'', ``Maxwell stratum'' and ``caustic'' divisors introduced in [11] via the set of standard generators of the rational Picard group. This generalizes the result of [11], where the expansion of the full discriminant divisor (which is a linear combination of the classes mentioned above) was obtained. In the second part of the paper we derive a formula that relates two Hodge classes in the rat…
Using the developed deformation theory on moduli spaces of quadratic differentials we derive variational formulas for objects associated with generalized $SL(2)$ Hitchin's spectral covers: Prym matrix, Prym bidifferential, Hodge and Prym tau-functions. The resulting formulas are antisymmetric versions of Donagi-Markman residue formula. The second variation of the Prym matrix is a natural analogy to the expression previously derived for the period matrix of the $GL(n)$ spectral cover.
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…
Using the embedding of the moduli space of generalized GL(n) Hitchin's spectral covers to the moduli space of meromorphic abelian differentials we study the variational formulae of the period matrix, the canonical bidifferential, the prime form and the Bergman tau function. This leads to residue formulae which generalize the Donagi-Markman formula for variations of the period matrix. Computation of second derivatives of the period matrix reproduces the formula derived by Baraglia and Zhenxi Huang using the framework of topological recursion.
In \cite{kim11} we have generalized a tangency condition in the Treibich-Verdier theory \cite{trei89,tv90,trei97} about elliptic solitons to a Hitchin system. As an application of this generalization, we will define, so-called, Hitchin hyperelliptic tangential covers and show the finiteness of them among all Hitchin tangential covers.
It is shown that the map from the Jacobian of the spectral curve to the moduli of stable bundles of rank 2 is generically simply branched along an irreducible divisor. This observation falsifies the key step in the "abelianization of the SU(2) WZW connection" presented in a recent paper [Yoshida, Annals 2006]
Visible Lagrangians for Hitchin Systems and Pillowcase Covers
本文研究了希奇金系统中可见拉格朗日流形的构造及其镜像对偶膜的关联，揭示了枕形覆盖与黎曼表面的关系。
The Hitchin morphism for K-trivial varieties
研究高维变种的希奇金映射，证明对于一类称为r-小的变种，希奇金映射的集论像等于谱基，验证了陈和诺的猜想增强版本，包含K-平凡变种。
Stratifying moduli spaces of Higgs bundles and the Hitchin morphism
研究Higgs束模空间的分层结构，并确定希奇金映射在特定条件下的图像。
On the generic fibers and true base of parabolic $\mathrm{SO}_{2n}$-Hitchin systems
本文证实了抛物型 SO(2n)-Hitchin 系统的物理猜想，证明 Hitchin 映射通过同构于仿射空间的有限覆盖分解，并引入剩余幂零 Hitchin 系统分析一般纤维，揭示其自对偶性与特殊幂零轨道的联系。
We study the locus of the moduli space of Higgs bundles on a curve given by those Higgs bundles obtained by pushforward under an unramified cover. We equip these loci with a hyperholomorphic bundle so that they can be viewed as BBB-branes, and we introduce corresponding BAA-branes which can be described via Hecke modifications. We then show how these branes are naturally dual via explicit Fourier--Mukai transform, where we recall that the structure group $\mathrm{GL}(n,\mathbb{C})$ is Langlands self dual. It is noteworthy that these branes lie over the singular locus of the Hitchin fibration. …
Let $φ\colon X\rightarrow Y$ be a degree two Galois cover of smooth curves over a local field $F$ of odd characteristic. Assuming that $Y$ has good reduction, we describe a semi-stability criterion for the curve $X$, using the data of the branch locus of the covering $φ$. In the case that $X$ has semi-stable reduction, we describe the dual graph of the minimal regular model of $X$ over $F.$ We do this by adopting the notion of cluster picture defined for hyperelliptic curves for the case where $Y$ is not necessarily a rational curve. Using these results, we describe the variation of the p-adic…
We compute the canonical model of the cover of Shimura curves $X_0(2) \to X(1)$ for the cubic field of discriminant 13^2 described at the end of Elkies' paper "Shimura curves for level 3 subgroups of the (2,3,7) triangle group". Last, we list the coordinates of some rational CM points on X(1).
Invariant vector bundles and Hitchin systems
本文研究亏格至少为2的光滑复射影曲线伽罗瓦覆盖上半稳定不变量向量丛的模空间，分类其连通分支，并确定相应希钦系统在光滑情形下的纤维结构。
In this paper, we explore the structure of the Hitchin morphism for higher dimensional varieties. We show that the Hitchin morphism factors through a closed subscheme of the Hitchin base, which is in general a non-linear subspace of lower dimension. We conjecture that the resulting morphism, which we call the spectral data morphism, is surjective. In the course of the proof, we establish connections between the Hitchin morphisms for higher dimensional varieties, the invariant theory of the commuting schemes, and Weyl's polarization theorem. We use the factorization of the Hitchin morphism to c…
We define and parametrise so-called $\mathfrak{sl}(2)$-type fibres of the $\mathsf{Sp}(2n,\mathbb{C})$- and $\mathsf{SO}(2n+1,\mathbb{C})$-Hitchin system. These are (singular) Hitchin fibres, where the spectral curve induces a two-sheeted covering of a second Riemann surface $Y$. This identifies the $\mathfrak{sl}(2)$-type Hitchin fibres with fibres of an $\mathsf{SL}(2,\mathbb{C})$- respectively $\mathsf{PSL}(2,\mathbb{C})$-Hitchin map on $Y$. We give a stratification of these singular spaces by semi-abelian spectral data, study their irreducible components and obtain a global description of …
The Hitchin Fibration for Symmetric Pairs
本研究引入并刻画对称空间希钦纤维化的正则商，针对特定GL2n空间给出谱覆盖的不变量论方法，研究拟分裂对的正则中心化子群概形及相关群概形的伽罗瓦描述，推导希格斯丛与对应希钦系统的若干基本结论。
Compactifying the rank two Hitchin system via spectral data on semistable curves
本文通过二次多重尺度微分空间，利用带基点稳定曲线的紧化雅可比簇，提供了秩二 Hitchin 系统的紧化，并将经典的谱对应扩展到了紧化的 Hitchin 系统。
We study mirror symmetry on the singular locus of the Hitchin system at two levels. Firstly, by covering it by (supports of) $(BBB)$-branes, corresponding to Higgs bundles reducing their structure group to the Levi subgroup of some parabolic subgroup $\mathrm{P}$, whose conjectural dual $(BAA)$-branes we describe. Heuristically speaking, the latter are given by Higgs bundles reducing their structure group to the unipotent radical of $\mathrm{P}$. Secondly, when $\mathrm{P}$ is a Borel subgroup, we are able to construct a family of hyperholomorphic bundles on the $(BBB)$-brane, and study the va…
Cyclic Spectral Data
本文研究特征零代数闭域上光滑簇的循环谱数据，证明非零循环谱数据的Hitchin纤维非空，并构造Hitchin态射截面及循环谱覆盖的平坦族。
The configuration space of $n$ marked points on the complex plane is considered. We investigate a decomposition of this space by so-called Gauss-skizze i.e. a class of graphs being forests, introduced by Gauss. It is proved that this decomposition is a semi-algebraic topological stratification. It also forms a cell decomposition of the configuration space of $n$ marked points. Moreover, we prove that classical tools from deformation theory, ruled by a Maurer--Cartan equation, can be used only locally for Gauss-skizze. We prove that the deformation of the Gauss-skizze is governed by a Hamilton-…
We consider the moduli space of stable principal G-bundles over a compact Riemann surface C of genus >1, with G a reductive algebraic group. We explicitly construct a map F from the generic fibre of the Hitchin map to a generalized Prym variety associated to a suitable Galois covering of C. The map F has finite fibres. In case G=PGl(2) one can check that the generic fibre of F is a principal homogeneous space with respect to a product of 2d-2 copies of Z/2Z where d is the degree of the canonical bundle over C. However in case the Dynkin diagram of G does not contain components of type $B_{n}$ …
On the spectral variety for rank two Higgs bundles
本文针对射影流形上的秩二希格斯丛，通过将谱簇显式构造为底空间的双重分支覆盖得到有限 Cohen-Macaulay 化，从而证实 Chen-Ngô 猜想，并进一步构造希钦截面以研究特征簇刚性与 Milnor-Wood 型不等式推广。
Conductor-Discriminant Inequality for Tamely Ramified Cyclic Covers I
本文证明了离散赋值域上Z/n-覆盖的P^1的导子-判别式不等式，推广了超椭圆曲线情形，并加强了Kohls关于Jacobian导子指数的结果。
One can associate to many of the well known algebraically integrable systems of Jacobians (generalized Hitchin systems, Sklyanin) a ruled surface which encodes much of its geometry. If one looks at the classification of such surfaces, there is one case of a ruled surface that does not seem to be covered. This is the case of projective bundle associated to the first jet bundle of a topologically nontrivial line bundle. We give the integrable system corresponding to this surface; it turns out to be a deformation of the Hitchin system.
We study real double covers of $\mathbb P^1\times\mathbb P^2$ branched over a $(2,2)$-divisor, which have the structure of a conic bundle threefold with smooth quartic discriminant curve via the second projection. In each isotopy class of smooth plane quartics, we construct examples where the total space of the conic bundle is rational. For five of the six isotopy classes we construct $\mathbb C$-rational examples that have obstructions to rationality over $\mathbb R$, and for the sixth class, we show that the models we consider are all rational. Moreover, for three of the five classes with ir…
Comptage de fibrés de Hitchin pour le groupe $\mathrm{SL}(n)$
该研究在有限域光滑射影曲线框架下，给出了$\boldsymbol{\text{SL}(n)}$群的稳定$D$-扭曲Hitchin丛数目的循环覆叠表达公式，并结合已有工作推导了对应模空间的多类不变量闭式公式，核心工具为自守诱导基本引理与相对上同调支撑定理。
Conductor-Discriminant Inequality for Tamely Ramified Cyclic Covers II
本文为温和分歧循环覆盖建立 conductor-discriminant 不等式，证明其贡献量非负性并刻画负值情形为有理双重点，推广了 Ogg、Saito 等人的工作。
The notion of a universal building associated with a point in the Hitchin base is introduced. This is a building equipped with a harmonic map from a Riemann surface that is initial among harmonic maps which induce the given cameral cover of the Riemann surface. In the rank one case, the universal building is the leaf space of the quadratic differential defining the point in the Hitchin base. The main conjectures of this paper are: (1) the universal building always exists; (2) the harmonic map to the universal building controls the asymptotics of the Riemann-Hilbert correspondence and the non-a…
This is a survey of various results about spectral covers and their relationship to Higgs bundles. To a G-principal Higgs bundle on a variety S corresponds a cameral cover \widetilde{S} of S (a W-Galois cover, where W is the Weyl group of G) together with a sheaf on \widetilde{S} which in simple cases is a line bundle, and is W-equivariant up to certain twists and shifts. Various other types of spectral covers, depending on the choice of a representation or weight of G, arise as associated objects of \widetilde{S}. We focus on the decomposition of the Picards of these spectral covers into Prym…
Basecondary polytopes
本文引入basecondary polytopes类多胞体，可统一研究次级多胞体、基多胞体及牛顿多胞体等，还涵盖无法归约为传统A-判别式与次级多胞体的Lyashko–Looijenga映射判别式，为相关几何研究提供统一框架。
Fourier-Mukai transform for fine compactified Prym varieties
本文针对由平面奇异谱曲线定义的紧化 Prym 簇构造 Fourier–Mukai 变换，建立其与等变导出范畴的等价，从而得到稠密一类奇异谱曲线上 SL(n,C) 与 PGL(n,C) Hitchin 纤维的导出等价。
We study the arithmetic of Enriques surfaces whose universal covers are singular K3 surfaces. If a singular K3 surface X has discriminant d, then it has a model over the ring class field d. Our main theorem is that the same holds true for any Enriques quotient of X. It is based on a study on Neron-Severi groups of singular K3 surfaces. We also comment on Galois actions on divisors of Enriques surfaces.
Sparsity of rational points on torsion level covers of Hilbert modular varieties
本文证明 Hilbert 模簇挠点层覆盖上有理点随水平范数增大而稀疏，给出对数典范度下界，并推广了几何挠点定理。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。