Wild nonabelian Hodge theory on curves

Hyperkahler manifolds and nonabelian Hodge theory of (irregular) curves

Short survey based on talk given at the Institut Henri Poincare January 17th 2012, during program on surface groups. The aim was to describe some background results before describing in detail (in subsequent talks) the results of [Boa11c] related to wild character varieties and irregular mapping class groups.

Opers versus nonabelian Hodge

For a complex simple simply connected Lie group $G$, and a compact Riemann surface $C$, we consider two sorts of families of flat $G$-connections over $C$. Each family is determined by a point ${\mathbf u}$ of the base of Hitchin's integrable system for $(G,C)$. One family $\nabla_{\hbar,{\mathbf u}}$ consists of $G$-opers, and depends on $\hbar \in {\mathbb C}^\times$. The other family $\nabla_{R,ζ,{\mathbf u}}$ is built from solutions of Hitchin's equations, and depends on $ζ\in {\mathbb C}^\times, R \in {\mathbb R}^+$. We show that in the scaling limit $R \to 0$, $ζ= \hbar R$, we have $\nab…

Opers与非阿贝尔Hodge：数值研究

Opers and nonabelian Hodge: numerical studies

本文通过在复平面上多项式全纯微分环境下进行数值实验，测试了关于opers单值性映射、非阿贝尔Hodge对应及Hitchin超凯勒度量的猜想，实验结果总体支持该猜想。

The Kapustin-Witten equations and nonabelian Hodge theory

Arising from a topological twist of $\mathcal{N} = 4$ super Yang-Mills theory are the Kapustin-Witten equations, a family of gauge-theoretic equations on a four-manifold parametrized by $t\in\mathbb{P}^1$. The parameter corresponds to a linear combination of two super charges in the twist. When $t=0$ and the four-manifold is a compact Kähler surface, the equations become the Simpson equations, which was originally studied by Hitchin on a compact Riemann surface, as demonstrated independently in works of Nakajima and the third-named author. At the same time, there is a notion of $λ$-connection …

Wild Hodge theory and Hitchin-Kobayashi correspondence [after T. Mochizuki]

T. Mochizuki constructs a theory of variations of wild Hodge structure for which the underlying flat connection can have irregular singularities at infinity. He extends in this way the correspondence of Corlette and Simpson between irreducible flat bundles and stables Higgs bundles, taking into account objects with irregular singularities. As an application, he proves a conjecture of Kashiwara concerning a generalization of the Hard Lefschetz theorem when the coefficients are the de Rham complex of a simple holonomic D-module on smooth complex projective variety.

非阿贝尔Hodge类型的Lagrange对应与平移扭曲结构

Lagrangian correspondences of nonabelian Hodge type and shifted twistor structures

本文建立平坦与Higgs完美复形的导出栈之间的Lagrange对应，并证明光滑射影簇上的解析Deligne-Hitchin-Simpson模栈具有规范平移预扭曲结构。

Global Weyl groups and a new theory of multiplicative quiver varieties

In previous work a relation between a large class of Kac-Moody algebras and meromorphic connections on global curves was established---notably the Weyl group gives isomorphisms between different moduli spaces of connections, and the root system is also seen to play a role. This involved a modular interpretation of many Nakajima quiver varieties, as moduli spaces of connections, whenever the underlying graph was a complete k-partite graph (or more generally a supernova graph). However in the isomonodromy story, or wild nonabelian Hodge theory, slightly larger moduli spaces of connections are co…

复向量丛上的联络、度规和Higgs场

Connections, metrics and Higgs fields on complex fiber bundles

通过曲率表示全纯纤维化的扩张类，推广Atiyah工作；得到Weil定理的非线性模拟；建立Kähler型约化非线性平坦丛到非线性Higgs丛的忠实函子；定义非线性调和丛并证明非Abel Hodge结构的变分在秩一和半单情形下为其特例。

曲线上的亚纯仿射Weyl群Higgs挠子与带滤过的Stokes G-局部系统

Meromorphic Parahoric Higgs Torsors and Filtered Stokes G-local Systems on Curves

本文在Boalch提出的“非常好”条件下建立曲线上主G-丛的野非阿贝尔Hodge对应，证实其猜想，给出两类一一对应，且G为GLₙ(ℂ)时结果退化为已有结论。

抛物型 SL(n,C)-Higgs 丛的共形极限

Conformal limits for parabolic SL(n,C)-Higgs bundles

本文将 Higgs 丛与全纯联络的共形极限对应推广到抛物型情形，证明在温和泛型条件下共形极限存在并给出保持自然分层的全纯截面，同时构造了允许留数特征值变化的抛物型 Higgs 丛模空间，并揭示了抛物型情形中非阿贝尔 Hodge 对应与共形极限的新现象。

Higgs 丛、调和映射与折曲面

Higgs bundles, harmonic maps, and pleated surfaces

本文借助非 Abel Hodge 对应与调和映射高能量退化，建立 SU(2) 自对偶方程解极限构型和双曲三维等变折曲面剪切—弯曲参数的渐近对应，并给出模空间紧化等应用。