内窥分解与Hausel-Thaddeus猜想

Motivic Integration on the Hitchin Fibration

We prove that the moduli spaces of twisted $\mathrm{SL}_n$ and $\mathrm{PGL}_n$-Higgs bundles on a smooth projective curve have the same (stringy) class in the Grothendieck ring of rational Chow motives. On the level of Hodge numbers this was conjectured by Hausel and Thaddeus, and recently proven by Groechenig, Ziegler and the second author. To adapt their argument, which relies on p-adic integration, we use a version of motivic integration with values in rational Chow motives and the geometry of Néron models to evaluate such integrals on Hitchin fibers.

Complex K-theory of moduli spaces of Higgs bundles

We establish an isomorphism of complex $K$-theory of the moduli space $\check{\mathcal{M}}$ of $``SL_n"$-Higgs bundles of degree $d$ and rank $n$ (in the sense of Hausel--Thaddeus) and twisted complex $K$-theory of the orbifold $\hat{\mathcal{M}}$ of $PGL_n$-Higgs bundles of degree $e$, where $(n,d)=(n,e)=1$. Along the way we prove the vanishing of torsion for $H^*(\check{\mathcal{M}})$ and certain twisted complex $K$-theory groups of $\hat{\mathcal{M}}$. We also extend Arinkin's autoduality of compactified Jacobian to a derived equivalence between $SL_n$ and $PGL_n$-Hitchin systems over the e…

Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration

We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type $\operatorname{SL}_n$ and $\operatorname{PGL}_n$. More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises p-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore we prove for $d$ coprime to $n$, that the number of rank $n$ Higgs bundles of deg…

Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces

Let M be a moduli space of stable sheaves on a K3 or Abelian surface S. We express the class of the diagonal in the cartesian square of M in terms of the Chern classes of a universal sheaf. Consequently, we obtain generators of the cohomology ring of M. When S is a K3 and M is the Hilbert scheme of length n subschemes, this set of generators is sufficiently small in the sense that there aren't any relations among them in the stable cohomology ring. When S is the cotangent bundle of a Riemann surface, we recover the result of T. Hausel and M. Thaddeus: The cohomology ring of the moduli spaces o…

Unramified covers and branes on the Hitchin system

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. …

曲线上$\mathrm{SL}_n$-希格斯丛模空间的交截上同调

On the intersection cohomology of the moduli of $\mathrm{SL}_n$-Higgs bundles on a curve

本文利用Ngô型支撑不等式研究曲线上任意次数SL_n-希格斯丛模空间的交截上同调，证明了奇异情形下的支撑定理及Hausel-Thaddeus拓扑镜像对称猜想的一个版本，并推广了Harder-Narasimhan定理。

希格斯丛的拟BPS范畴的拓扑K-理论

Topological K-theory of quasi-BPS categories for Higgs bundles

该研究验证了希格斯丛BPS范畴间的猜想等价关系在拓扑K-理论层面成立，同时证明BPS范畴的拓扑K-理论同构于半稳定希格斯丛模空间的BPS上同调，为相关几何朗兰兹与镜像对称研究提供了拓扑K-理论层面的支撑。

$SL_n$ 与 $PGL_n$ Higgs 丛的非阿基米德拓扑镜像对称

Non-archimedean topological mirror symmetry for $SL_n$ and $PGL_n$ Higgs bundles

本文将非阿基米德拓扑镜像对称方法推广到非互素情形，证明任意次数下 $SL_n$ 与 $PGL_n$ Higgs 丛模空间的 gerbe 扭曲 $p$-进体积相等，并建立其与相交上同调及 $BPS$ 上同调的联系。

希格斯丛与Gopakumar-Vafa不变量的上同调χ-独立性

Cohomological $χ$-independence for Higgs bundles and Gopakumar-Vafa invariants

本文证明了局部曲线Gopakumar-Vafa不变量的Toda χ-独立性猜想，并引入不依赖χ的希格斯丛模空间BPS上同调，将Hausel-Thaddeus猜想推广到混合Hodge模同构层面且无需互素假设。

Higgs 丛的拟 BPS 范畴

Quasi-BPS categories for Higgs bundles

本文为扭曲 Higgs 丛引入拟 BPS 范畴，提出交换 Euler 示性数与权的对称猜想，并在秩二亏格零情形及高亏格稳定模空间导出等价中取得证明。

通过极限范畴的多贝尔几何朗兰兹猜想

The Dolbeault geometric Langlands conjecture via limit categories

该研究引入光滑栈余切栈的极限范畴，发展其理论并探讨与 D -模范畴关系，提出多贝尔几何朗兰兹猜想精确表述，证明极限范畴半正交分解，构造赫克算子，预期其蕴含半稳定希格斯丛 BPS 范畴等价，是拓扑镜像对称猜想的范畴版本。

动机镜像对称与Higgs丛在任意特征下的χ独立性

Motivic mirror symmetry and $χ$-independence for Higgs bundles in arbitrary characteristic

本文证明了SL_n和PGL_n-Higgs丛在任意特征下的动机同构性，并建立了GL_n-Higgs丛的动机χ独立性。

对称栈的上同调

Cohomology of symmetric stacks

本文在存在好模空间和逐点正交切空间的条件下，构造了光滑栈、$0$-移位辛栈和$(-1)$-移位辛栈的上同调分解，定义了BPS上同调，并推广了PBW型定理与拓扑镜像对称及Langlands对偶猜想。