Opers与非阿贝尔Hodge：数值研究

Nonabelian Hodge theory and vector valued modular forms

We examine the relationship between nonabelian Hodge theory for Riemann surfaces and the theory of vector valued modular forms. In particular, we explain how one might use this relationship to prove a conjectural three-term inequality on the weights of free bases of vector valued modular forms associated to complex, finite dimensional, irreducible representations of the modular group. This conjecture is known for irreducible unitary representations and for all irreducible representations of dimension at most 12. We prove new instances of the three-term inequality for certain nonunitary represe…

Nonabelian Hodge Theory for klt spaces and descent theorems for vector bundles

We generalise Simpson's nonabelian Hodge correspondence to the context of projective varieties with klt singularities. The proof relies on a descent theorem for numerically flat vector bundles along birational morphisms. In its simplest form, this theorem asserts that given any klt variety X and any resolution of singularities, then any vector bundle on the resolution that appears to come from X numerically, does indeed come from X. Furthermore and of independent interest, a new restriction theorem for semistable Higgs sheaves defined on the smooth locus of a normal, projective variety is esta…

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 …

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…

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。

霍奇猜想的狄拉克-多博尔特算子方法

The Dirac-Dolbeault Operator Approach to the Hodge Conjecture

该研究利用Dirac-Dolbeault算子相关的Green函数表示与Nash-Moser广义反函数定理，在特定假设下证明复射影流形满足条件的复子流形存在性，进而证明复射影流形的Hodge猜想。

The multiplicity of eigenvalues of the Hodge Laplacian on 5-dimensional compact manifolds

We study multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact two-forms, has nonzero eigenvalues of multiplicity two. The proof is based on the fact that Hodge Laplacian restricted to the subspace of co-exact two-forms is minus the square of the Beltrami operator, a first-order operator. We prove that for generic metrics the spectrum of the Beltrami operator is simple. Because the Beltrami operator in this setting is …

Hodge-Dirac 算子的谱度量与 Einstein 泛函

Spectral Metric and Einstein Functionals for Hodge-Dirac operator

本文研究 Hodge-Dirac 算子的度量与 Einstein 泛函，证明其与旋量流形上 Dirac 算子的泛函仅差数值因子，并证明谱三元组谱闭且无挠。