Xin Tong
In this paper, we give the introduction to the Hodge-Iwasawa Theory introduced by the author. After that we will give some well-defined extensions to the already shaped framework established in our previous work.
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$\infty$-Categorical Approaches to Hodge-Iwasawa Theory II: $\infty$-Categorical and Derived Hodge-Iwasawa Modules
本文探讨$\infty$-范畴方法在Hodge-Iwasawa理论中的应用，旨在通过形变Hodge理论构造，统一$p$-进Hodge理论与$p$-进Iwasawa理论，并研究相应的$\infty$-范畴导出Hodge-Iwasawa模。
Arithmetic Hodge-Iwasawa Moduli Stacks
本文在已有Hodge-Iwasawa理论基础上构造G-丛的算术模叠，用以连贯参数化Hodge-Iwasawa结构。
In this paper we study the corresponding categories and the corresponding cohomologies of the Hodge-Iwasawa modules we developed in our series papers on Hodge-Iwasawa theory. The corresponding cohomologies will be essential in the corresponding development of the contact with the corresponding Iwasawa theoretic consideration, while they are as well very crucial in the corresponding study of the corresponding deformations of local systems over general analytic spaces. We contact with some applications in analytic geometry and arithmetic geometry which all have their own interests and deserve fu…
In this paper, we are going to establish a simultaneous generalization of the relative Iwasawa theory proposed by Kedlaya-Pottharst and the relative $p$-adic Hodge theory after Kedlaya-Liu. We call this Hodge-Iwasawa theory in the sense that one could apply the theory to study noncommutative Iwasawa cohomology and noncommutative Iwasawa theories in families and meanwhile one could apply the theory to study the deformation theory of étale local systems or families of representations of fundamental groups or the equivariant constructible $p$-adic sheaves, with more sophisticated point of view co…
We continue our study on the Hodge-Iwasawa theory which is a continuation of our previous work on Hodge-Iwasawa theory, which is aimed at higher dimensional deformation of higher dimensional Hodge structures over general analytic spaces or adic spaces. We still follow closely the approaches of Kedlaya-Liu to study our Frobenius modules over the different kinds of period rings including more generalized perfect period rings and the corresponding imperfect period rings. It is desirable that one can globalize the deformed information in order to organize the deformed period rings into the corresp…
We propose a new approach to the Mirror Symmetry Conjecture in a form suitable to possibly non-Kähler compact complex manifolds whose canonical bundle is trivial. We apply our methods by proving that the Iwasawa manifold $X$, a well-known non-Kähler compact complex manifold of dimension $3$, is its own mirror dual to the extent that its Gauduchon cone, replacing the classical Kähler cone that is empty in this case, corresponds to what we call the local universal family of essential deformations of $X$. These are obtained by removing from the Kuranishi family the two "superfluous" dimensions of…
One of the most mysterious aspects of Saito's theory of Hodge modules are the Hodge and weight filtrations that accompany the pushforward of a Hodge module under an open embedding. In this paper we consider the open embedding in a product of complementary Grassmannians given by pairs of transverse subspaces. The push-forward of the structure sheaf under this open embedding is an important Hodge module from the viewpoint of geometric representation theory and homological knot invariants. We compute the associated graded of this push-forward with respect to the induced Hodge filtration as well a…
Analytic Geometry and Hodge-Frobenius Structure
本文研究高维p进解析几何中的Frobenius结构及相关p进泛函分析，依托既有研究纲领与方法探究广义p进Hodge结构形变，为相关上同调、岩泽理论及非交换玉河数猜想研究奠基。
The subject of the present work is the de Rham part of non-commutative Hodge structures on the periodic cyclic homology of differential graded algebras and categories. We discuss explicit formulas for the corresponding connection on the periodic cyclic homology viewed as a bundle over the punctured formal disk. Our main result says that for the category of matrix factorizations of a polynomial the formulas reproduce, up to a certain shift, a well-known connection on the associated twisted de Rham cohomology which plays a central role in the geometric approach to the Hodge theory of isolated si…
We write down a new "logarithmic" quasicoherent category $\operatorname{Qcoh}_{log}(U, X, D)$ attached to a smooth open algebraic variety $U$ with toroidal compactification $X$ and boundary divisor $D$. This is a (large) symmetric monoidal Abelian category, which we argue can be thought of as the categorical substrate for logarithmic Hodge theory of $U$. We show that its Hochschild homology theory coincides with the theory of log-forms on $X$ with logarithmic structure induced by $D$, and in particular, that the noncommutative Hodge-to de Rham sequence on $\operatorname{Qcoh}_{log}(U, X, D)$ r…
Cyclic covers: Hodge theory and categorical Torelli theorems
本文研究带矩形Lefschetz分解基簇上的分歧循环覆盖，定义包含$\mathrm{D}^{\mathrm{b}}(Z)$的等变子范畴$\mathcal{A}_Z$，分解Kuznetsov分量，并推出低次数素Fano三维簇的范畴Torelli定理。
Infinitesimal Torelli problems for special Gushel-Mukai and related Fano threefolds: Hodge theoretical and categorical perspectives
本文研究特殊Gushel-Mukai及相关Fano三维簇的无穷小Torelli问题，证明特殊Gushel-Mukai三维簇的无穷小周期映射不变部分单射，并给出周期映射微分核的几何描述。
Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration
本文否证了 Kontsevich 关于光滑与 proper DG 范畴的广义 Hodge-to-de Rham 退化猜想，构造了特征零域上超迹非零的 10 维 A∞-代数，并给出不具有光滑范畴紧化及不承认范畴化奇点消解的 DG 范畴反例，否定回答了 Toën 的问题。
Mixed Hodge modules on stacks
本文借助混合 Hodge 模的 ∞-范畴增强，将其连同六运算与权推广到代数栈，并证明 Drew 的 motivic Hodge 模范畴全忠实嵌入且像为几何来源的混合 Hodge 模。
$\infty$-Categorical Generalized Langlands Program I: Mixed-Parity Modules and Sheaves
本研究在小v-栈框架下构建混合奇偶性场景的p进霍奇理论基础，研究对应模v-栈，为广义朗兰兹纲领提供支撑，并给出Moduli_G上理论的p进上同调化实现。
In this paper, we present an explicit cyclic minimal $A_\infty$ model for the category of matrix factorizations $\MF(W)$ of an isolated hypersurface singularity. The key observation is to use Kontsevich's deformation quantization technique. Pushing this idea further, we use the Tsygan formality map to obtain a comparison theorem that the categorical Variation of Semi-infinite Hodge Structure of $\MF(W)$ is isomorphic to Saito's original geometric construction in primitive form theory. An immediate corollary of this comparison result is that the analogue of Caldararu's conjecture holds for the …
Finite-Node Perverse Schobers and Corrected Extensions for Conifold Degenerations
本文研究具有有限普通双点的单参数 Conifold 褶皱，提出有限节点 bulk/localized-sector 形式化框架，识别修正有限节点 perverse 扩展的第一层范畴结构，并构建混合 Hodge 模包。
We define combinatorial counterparts to the geometric string vertices of Sen-Zwiebach and Costello-Zwiebach, which are certain closed subsets of the moduli spaces of curves. Our combinatorial vertices contain the same information as the geometric ones, are effectively computable, and act on the Hochschild chains of a cyclic $A_\infty$-algebra. This is the first in a series of two papers where we define enumerative invariants associated to a pair consisting of a cyclic $A_\infty$-algebra and a splitting of the Hodge filtration on its cyclic homology. These invariants conjecturally generalize th…
The $p$-adic monodromy theorem over algebraic-affinoid algebras
本文研究代数仿射代数参数化的Fargues-Fontaine曲线上$G_K$-等变向量丛族的$p$-进霍奇理论性质，定义相关表示概念并证明$p$-进单值化定理，推广了Berger-Colmez的工作，还在去掉自由性条件下给出了等变线丛族的分类。
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