LG/CY correspondence: the state space isomorphism

Bi-algebraicity in the rank one Riemann--Hilbert correspondence via o-minimality

For a smooth, projective, complex algebraic variety $X$, the Riemann--Hilbert correspondence establishes a complex analytic isomorphism between the `Betti moduli space' of rank $n$ local systems on $X^\mathrm{an}$ and the `de Rham moduli space' of rank $n$ vector bundles with flat connection on $X$. In the rank one case, C. Simpson precisely characterizes the subvarieties of these moduli spaces that are `bi-algebraic' for this typically transcendental, analytic isomorphism. In this short note, we give a new proof of this characterization of Simpson, using methods from o-minimal geometry. We ad…

On correspondences of a K3 surface with itself. III

Let $X$ be a K3 surface, and $H$ its primitive polarization of the degree $H^2=2rs$, $r,s\ge 1$. The moduli space of sheaves over $X$ with the isotropic Mukai vector $(r,H,s)$ is again a K3 surface, $Y$. In math.AG/0206158, math.AG/0304415 and math.AG/0307355 (in general) we gave necessary and sufficient conditions in terms of Picard lattice $N(X)$ of $X$ when $Y$ is isomorphic to $X$, under the additional condition $H\cdot N(X)=\bz$. Here we show that these conditions imply existence of an isomorphism between $Y$ and $X$ which is a composition of some universal isomorphisms between moduli of …

Shimura 簇的几何 Jacquet--Langlands 对应

A geometric Jacquet--Langlands correspondence for Shimura varieties

本文提出并证明了一个关于不同 Shimura 数据（底层群互为纯内形式）的 Igusa 栈间异种同构的猜想，结合谱作用关联上同调，给出 Jacquet--Langlands 对应的几何体现。

关于$\mathbb{B}_{\mathrm{dR}}^+$上的光滑adic空间和sheafified $p$-adic Riemann--Hilbert对应

On smooth adic spaces over $\mathbb{B}_{\mathrm{dR}}^+$ and sheafified $p$-adic Riemann--Hilbert correspondence

本文研究了$\mathbb{B}_{\mathrm{dR}}^+$上的光滑adic空间，并构造了sheafified $p$-adic Riemann--Hilbert对应，推广了Heuer的$p$-adic Simpson对应工作。

The Euclid-Fourier-Mukai algorithm for elliptic surfaces

We describe the birational correspondences, induced by the Fourier-Mukai functor, between moduli spaces of semistable sheaves on elliptic surfaces with sections, using the notion of $P$-stability in the derived category. We give explicit conditions to determine whether these correspondences are isomorphisms. This is indeed not true in general and we describe the cases where the birational maps are Mukai flops. Moreover, this construction provides examples of new compactifications of the moduli spaces of vector bundles via sheaves with torsion and via complexes. We finally get for any fixed dim…

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。

簇堆叠包的crepant变换对应关系

Crepant Transformation Correspondence For Toric Stack Bundles

本文通过构造辛变换和 Fourier-Mukai 同构，研究了簇堆叠包在 crepant 墙交叉下的 Gromov-Witten 理论对应关系。

用模空间表述的小$p$进Simpson对应

The small $p$-adic Simpson correspondence in terms of moduli spaces

本研究针对$\text{Q}_p$的perfectoid扩张上的刚性空间，基于Bhatt-Lurie的Hodge-Tate栈构造两类模栈的同构，改进了小$p$进Simpson对应，并在算术设定下证明了类似结论。

Comparaison de la cohomologie des tours de Lubin-Tate et de Drinfeld et correspondance de Jacquet-Langlands geometrique

This article is the last one about the isomorphism between Lubin-Tate and Drinfeld towers. We prove the existence of an isomorphism between the compactly supported etale cohomology of the Lubin-Tate and Drinfeld towers, and more generally their equivariant cohomology complex. We also prove the existence of a geometric local Jacquet-Langlands correspondence between some equivariant rigid etale sheaves on Gross-Hopkins period space $\mathbb{P}^{n-1}$ and Drinfeld one $Ω$.

关于Grassmann空间同构的研究

On Isomorphisms of Grassmann Spaces

本文展示了Grassmann空间的同构，并揭示了它们与底层射影空间的共线性和嵌入之间的联系。

Isomorphisms of Moduli Spaces

We give infinitely many new isomorphisms between moduli spaces of bundles on local surfaces and on local Calabi--Yau threefolds.

Isomorphisms of Affine Plücker Spaces

All isomorphisms of Plücker spaces on affine spaces with dimensions $\geq 3$ arise from collineations of the underlying affine spaces.

等变拉格朗日对应与Teleman猜想

Equivariant Lagrangian correspondence and a conjecture of Teleman

本文通过将Fukaya的$A_\infty$三模构造推广到Borel空间，研究等变拉格朗日对应的Floer理论，证明等变导出Fukaya范畴与辛商空间导出Fukaya范畴拟同构，并利用等变阻碍导致的量子修正解决Teleman猜想。

Modular Calabi-Yau threefolds of level eight

In the studies on the modularity conjecture for rigid Calabi-Yau threefolds several examples with the unique level 8 cusp form were constructed. According to the Tate Conjecture correspondences inducing isomorphisms on the middle cohomologies should exist between these varieties. In the paper we construct several examples of such correspondences. In the constructions elliptic fibrations play a crucial role. In fact we show that all but three examples are in some sense built upon two modular curves from the Beauville list.

一种广义谱对应

A generalized spectral correspondence

本文研究代数曲线上任意秩层与扭曲对之间的广义谱对应，利用复合推拉投影公式提出迭代谱对应猜想，并通过 Galois 群本原性分类对一类 $\mathbb{P}^1$ 谱覆盖予以证明。