一种广义谱对应

On the action of algebraic correspondences on weight spectral sequences

In a work of T. Saito, the action of algebraic correspondences on the etale cohomology of varieties over local fields with semistable reduction is related to correspondences on smaller strata via weight spectral sequences. We give an intersection theoretic construction of these correspondences. Under a finiteness condition this enables us to compute them without involving the blow-ups of products, and prove their compatibility with compositions. These features are essential for the application to Shimura varieties.

约化谱曲线情形下 $\mathrm{GL}_2$ 的 Dolbeault 几何 Langlands 对应证明

A proof of Dolbeault geometric Langlands for $\mathrm{GL}_2$ with reduced spectral curves

本文证明了对 $\mathrm{GL}_2$ 在谱曲线约化的 Hitchin 基上的 Dolbeault 几何 Langlands 对应，引入极限范畴作为经典极限，并连接了范畴 Donaldson-Thomas 理论与几何 Langlands 对应。

The local monodromy as a generalized algebraic correspondence

In the paper we show that for a normal-crossings degeneration $Z$ over the ring of integers of a local field with $X$ as generic fibre, the local monodromy operator and its powers determine invariant cocycle classes under the decomposition group in the cohomology of the product $X \times X$. More precisely, they also define algebraic cycles on the special fibre of a resolution of $Z \times Z$. In the paper, we give an explicit description of these cycles for a degeneration with at worst triple points as singularities. These cycles explain geometrically the presence of poles on specific local f…

Twisted spectral correspondence and torus knots

Cohomological invariants of twisted wild character varieties as constructed by Boalch and Yamakawa are derived from enumerative Calabi-Yau geometry and refined Chern-Simons invariants of torus knots. Generalizing the untwisted case, the present approach is based on a spectral correspondence for meromorphic Higgs bundles with fixed conjugacy classes at the marked points. This construction is carried out for twisted wild character varieties associated to (l, kl-1) torus knots, providing a colored generalization of existing results of Hausel, Mereb and Wong as well as Shende, Treumann and Zaslow.

Grassmannian framed bundles and generalized parabolic structures

We build compact moduli spaces of Grassmannian framed bundles over a Riemann surface, essentially replacing a group by its bi-invariant compactification. We do this both in the algebraic and symplectic settings, and prove a Hitchin-Kobayashi correspondence between the two. The spaces are universal spaces for parabolic bundles, and the reduction to parabolic bundles commutes with the correspondence. An analogous correspondence is outlined for the generalized parabolic bundles of Bhosle.

On the Hodge-BGW correspondence

We establish an explicit relationship between the partition function of certain special cubic Hodge integrals and the generalized Brezin--Gross--Witten (BGW) partition function, which we refer to as the Hodge-BGW correspondence. As an application, we obtain an ELSV-like formula for generalized BGW correlators.

Painlevé 系统的秩三表示：III. Dolbeault 结构与谱对应

Rank three representations of Painleve systems: III. Dolbeault structure, spectral correspondence

本文证明 Painlevé 系统秩 2 与秩 3 表示在 Dolbeault 复结构下全纯辛同构，并结合 de Rham 描述确立相应模空间的超 Kähler 等距性。

The spectral $p$-adic Jacquet-Langlands correspondence and a question of Serre

We show that the completed Hecke algebra of $p$-adic modular forms is isomorphic to the completed Hecke algebra of continuous $p$-adic automorphic forms for the units of the quaternion algebra ramified at $p$ and $\infty$. This gives an affirmative answer to a question posed by Serre in a 1987 letter to Tate. The proof is geometric, and lifts a mod $p$ argument due to Serre: we evaluate modular forms by identifying a quaternionic double-coset with a fiber of the Hodge-Tate period map, and extend functions off of the double-coset using fake Hasse invariants. In particular, this gives a new proo…

对应与稳定同伦理论

Correspondences and stable homotopy theory

本文提出由一般范畴中的对称环对象生成对应与谱范畴的通用方法，并据此分别恢复稳定同伦理论 SH 与稳定动机同伦理论 SH(k)。

A Kobayashi-Hitchin correspondence for $I_\pm$-holomorphic bundles

In this paper, we introduce the notions of $α$-Hermitian-Einstein metric and $α$-stability for $I_\pm$-holomorphic vector bundles on bi-Hermitian manifolds. Moreover, we establish a Kobayashi-Hitchin correspondence for $I_\pm$-holomorphic vector bundles on bi-Hermitian manifolds. Examples of such vector bundles include generalized holomorphic bundles on generalized Kähler manifolds. We also show that $α$-stability of a vector bundle, in this sense, can depend on the parameter $α$.

相对谱对应：抛物Higgs丛与Deligne--Simpson问题

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。

关于$V$-扭曲Higgs包在穿孔曲线上的对数谱对应

Logarithmic spectral correspondence for $V$--twisted Higgs bundles on punctured curves

研究了穿孔曲线上的$V$-扭曲Higgs包的对数性质，通过Hecke紧致化方法分类了此类包，并扩展了ABK的谱对应理论到穿孔情况。

A note on p-adic Simpson correspondence

Given a proper smooth $p$-adic variety, we show a comparison theorem for the $p$-adic Simpson correspondence constructed by Faltings and Riemann-Hilbert correspondence constructed by Scholze. As an application we formulate a sufficient condition for $\overline{\mathbb Q}_p$-local system being de Rham. We study a $p$-adic analogue of Simpson's $\mathbb C^*$-action on the set of isomorphism classes of Higgs bundles and the corresponding Galois action on the set of isomorphism classes of generalized representations of the étale fundamental group.