相对上同调群的量子化

相对对数收敛上同调与相对刚性上同调III

Relative log convergent cohomology and relative rigid cohomology III

本文利用Kedlaya最近提出的关于过收敛$F$ -等晶的半稳定约化猜想，证明了相对刚性上同调的泛超收敛性.

Derived $δ$-Rings and Relative Prismatic Cohomology

We characterize the relative prismatic cohomology of Bhatt and Scholze by a universal property by endowing it with the additional structure of a ``derived $δ$-ring". This involves introducing an analogue of prismatic envelopes in the setting of filtered derived commutative rings and applying this construction to the Hodge filtration on infinitesimal cohomology. In particular, we recover relative prismatic cohomology from infinitesimal cohomology via a purely algebraic process.

Relative twisted homology and cohomology groups associated with Lauricella's $F_D$

We introduce relative twisted homology and cohomology groups associated with Euler type integrals of solutions to Lauricella's system $F_D(a,b,c)$ of hypergeometric differential equations. We define an intersection form between relative twisted homology groups and that between relative twisted cohomology groups, and show their compatibility. We prove that the relative twisted homology group is canonically isomorphic to the space of local solutions to $F_D(a,b,c)$ for any parameters $a,b,c$. Through this isomorphism, we study $\cF_D(a,b,c)$ by the relative twisted homology and cohomology groups…

On comparison between relative log de Rham-Witt cohomology and relative log crystalline cohomology

In this article, we prove the comparison theorem between the relative log de Rham-Witt cohomology and the relative log crystalline cohomology for a log smooth saturated morphism of fs log schemes satisfying certain condition. Our result covers the case where the base fs log scheme is etale locally log smooth over a scheme with trivial log structure or the case where the base fs log scheme is hollow, and so it generalizes the previously known results of Matsuue. In Appendix, we prove that our relative log de Rham-Witt complex and our comparison map are compatible with those of Hyodo-Kato.

Limiting mixed Hodge structures on the relative log de Rham cohomology groups of a projective semistable log smooth degeneration

We prove that the relative log de Rham cohomology groups of a projective semistable log smooth degeneration admit a natural \textit{limiting} mixed Hodge structure. More precisely, we construct a family of increasing filtrations and a family of nilpotent endomorphisms on the relative log de Rham cohomology groups and show that they satisfy a part of good properties of a nilpotnet orbit in several variables.

Relative quantum cohomology under birational transformations

We study how relative quantum cohomology, defined by Tseng--You and Fan--Wu--You, varies under birational transformations. For toric complete intersections with simple normal crossings divisors that contain the loci of indeterminacy, we prove that their respective relative $I$-functions can be directly identified. For toric complete intersections with smooth divisors, we prove that their respective relative $I$-functions are related by analytic continuation. We also study some connections with extremal transitions and FJRW theory.

Twisted GKZ hypergeometric functions and relative cohomology

We investigate the GKZ $A$-hypergeometric $\mathscr{D}$-modules, introduced by Gel'fand, Kapranov, and Zelevinskii, arising from cyclic covers of toric varieties and find its Riemann--Hilbert partner. This extends our earlier results in arXiv:1902.01536.

Hodge-Witt decomposition of relative crystalline cohomology

For a smooth and proper scheme over an artinian local ring with ordinary reduction over the perfect residue field we prove - under some general assumptions - that the relative de Rham-Witt spectral sequence degenerates and the relative crystalline cohomology, equipped with its display structure arising from the Nygaard complexes, has a Hodge-Witt decomposition into a direct sum of (suitably Tate-Twisted) multiplicative displays. As examples our main results include the cases of abelian schemes, complete intersections, varieties of K3 type and some Calabi-Yau $n$-folds.

相对德拉姆上同调有限性的一个 motive 证明

A motivic proof of the finiteness of relative de Rham cohomology

该研究针对Q上概形间光滑真态射X/S的相对德拉姆上同调群性质，用A1同伦理论替代霍奇理论与超越方法，快速证明其为基上向量丛。

$A$-hypergeometric systems and relative cohomology

We investigate the space of solutions to certain $A$-hypergeometric $\mathscr{D}$-modules, which were defined and studied by Gelfand, Kapranov, and Zelevinsky. We show that the solution space can be identified with certain relative cohomology group of the toric variety determined by $A$, which generalizes the results of Huang, Lian, Yau, and Zhu. As a corollary, we also prove the existence of rank one points for Calabi--Yau complete intersections in toric varieties.

相对胞腔空间的定向上同调与模分解

Oriented Cohomology and Motivic Decompositions of Relative Cellular Spaces

对于一个定向上同调理论A和一个相对胞腔空间X，我们将X的A-模分解为底空间的扭曲模的直和。我们还得到了X的A-上同调的相应分解。应用它们，可以计算A（X），其中X是迷向投射齐次簇，A表示代数K理论，动机上同调或代数协边MGL.

Relative $A_{\rm inf}$-cohomology

We construct a relative version of the $A_{\rm inf}$-cohomology theory developed by Bhatt-Morrow-Scholze and relate it to the prismatic theory of Bhatt-Scholze. The construction relies on the fiber product of topoi. As an application we show that there is an étale comparison of the $q$-crystalline pushforward after inverting $μ\in A_{\rm inf}$.

Relative Brauer Groups and étale cohomology

In this article, we construct a natural group homomorphism $$ ψ: \text{Br}(f)\to H^{1}_{et}(S, f_{*}\mathcal{O}_{X}^{\times}/\mathcal{O}_{S}^{\times})$$ for a faithful affine map $f: X \to S$ of noetherian schemes. Here $\text{Br}(f)$ denotes the relative Brauer group of $f.$ We also discuss the behavior of Brauer groups for a subintegral map. Furthermore, we prove a relative version of Kummer's exact sequence.

相对于δ环的棱柱上同调

Prismatic cohomology relative to $δ$-rings

本文发展了相对于δ环的棱柱和syntomic上同调，证明了相对棱柱上同调实际上不依赖棱柱结构，仅依赖于底层δ环。提供了三种等价的棱柱上同调定义，并应用于滤环的棱柱上同调理论。

The Belkale-Kumar cup product and relative Lie algebra cohomology

We study the Belkale-Kumar family of cup products on the cohomology of a generalized flag variety. We give an alternative construction of the family using relative Lie algebra cohomology, and in particular, identify the Belkale-Kumar cup product with a relative Lie algebra cohomology ring for every value of the parameter. As a consequence, we extend a fundamental disjointness result of Kostant to a family of Lie algebras. In an appendix, written jointly with Edward Richmond, we extend a Levi movability result of Belkale and Kumar to arbitrary parameters.

Overconvergent relative de Rham cohomology over the Fargues-Fontaine curve

We explain how to construct a cohomology theory on the category of separated quasi-compact smooth rigid spaces over $\mathbf{C}_p$ (or more general base fields), taking values in the category of vector bundles on the Fargues-Fontaine curve, which extends (in a suitable sense) Hyodo-Kato cohomology when the rigid space has a semi-stable proper formal model over the ring of integers of a finite extension of $\mathbf{Q}_p$. This cohomology theory factors through the category of rigid analytic motives of Ayoub.

代数几何与变形量子化中的导出运算中心

Derived operadic centers in algebraic geometry and deformation quantization

研究代数几何中霍赫希尔德上同调相关问题，通过将霍赫希尔德复形展示为\(\infty\)-运算中心，给出德利涅猜想典范解，为其配备通用\(\mathbb{E}_2\)-代数结构，并从运算形式模问题角度解释该结构。

多项式映射的相对上同调

Relative cohomology of polynomial mappings

设$F$是从$\mathbb{C}^n$到$\mathbb{C}^q$的多项式映射，其中$n>q$。通过引入一个特殊的纤维$F^{-1}(\infty)$“在无穷远处”及其上同调，研究了它的纤维及其相关上同调群的De Rham上同调.让我们在$\mathbb{C}[x_1,...,x_n]$上固定一个加权齐次度，其权重严格为正。无穷远处的纤维是$F$坐标函数首项的零点集。引入了$F$在无穷远处的上同调群$H^k(F^{-1}(\infty))$.这些群使我们能够计算$F$的所有其他上同调群。例如，如果无穷远纤维在原点有孤立奇点，则证明了$H^{n-q}(F^{-1}(\infty))$的每一个加权齐次基都是所有群$H^{n-q}(F^{-1}(y))$的基，也是$F$的$(n-q)^{th}$相对上同调群的基.此外，$H^{n-q}(F^{-1}(\infty))$的维数由$F$的整体Milnor数给出，该整体Milnor数仅取决于$F$的坐标函数的首项。

Relative Dolbeault cohomology

We review the notion of relative Dolbeault cohomology and prove that it is canonically isomorphic with the local (relative) cohomology of A. Grothendieck and M. Sato with coefficients in the sheaf of holomorphic forms. We deal with this cohomology from two viewpoints. One is the Cech theoretical approach, which is convenient to define such operations as the cup product and integration and leads to the study of local duality. Along the way we also establish some notable canonical isomorphisms among various cohomologies. The other is to regard it as the cohomology of a certain complex, which is …

Representation of relative sheaf cohomology

We study the cohomology theory of sheaf complexes for open embeddings of topological spaces and related subjects. The theory is situated in the intersection of the general Cech theory and the theory of derived categories. That is to say, on the one hand the cohomology is described as the relative cohomology of the sections of the sheaf complex, which appears naturally in the theory of Cech cohomology of sheaf complexes. On the other hand it is interpreted as the cohomology of a complex dual to the mapping cone of a certain morphism of complexes in the theory of derived categories. We prove a "…

相对上同调Picard-Fuchs方程的开弦镜像映射

Open string mirror maps from Picard- Fuchs equations on relative cohomology

在Lerche和Mayr的物理计算基础上，给出了一种计算非紧Calabi-Yaus的开弦镜像映射和超势的方法.它还表明，这些技术的明显的扩展到紧凑的情况下是不一致的。作为一个例子，局部CP^2情况用两种方法求解。

The Galois action and cohomology of a relative homology group of Fermat Curves

For an odd prime $p$ satisfying Vandiver's conjecture, we give explicit formulae for the action of the absolute Galois group $G_{\mathbb{Q}(ζ_p)}$ on the homology of the degree $p$ Fermat curve, building on work of Anderson. Further, we study the invariants and the first Galois cohomology group which are associated with obstructions to rational points on the Fermat curve.

相对（φ-Gamma）-模的整体应用I

Global applications of relative (phi-Gamma)-modules I

本文给出了p-adic dvr上的光滑真概型X和其上的p-幂挠局部系L，研究了L的局部相对（Phi-Gamma）-模的上同调所对应的层族及其上同调.作为应用，我们利用上述层的上同调，得到了X的几何泛纤维上取值于L的标准上同调群及其经典（Phi-Gamma）-模的描述.

SU(2,1)的相对李代数上同调与Picard曲面上的Eisenstein类

Relative Lie algebra cohomology of SU(2,1) and Eisenstein classes on Picard surfaces

该研究针对SU(2,1)对应的Picard曲面，通过分析SU(2,1)主系列的相对李代数上同调，构造了表示Eisenstein类的显式微分形式，为Bloch-Beilinson猜想相关的Hodge结构扩张类计算提供支撑。