Sur l'anneau de cohomologie du schéma de Hilbert de $\CC^2$
E. Vasserot
我们用对称群的代数中心来表示Hilbert概型的上同调环的乘积。对辛商奇点的可分解性给出了一个猜想。
We express the product of the cohomology ring of the Hilbert scheme in terms of the center of the algebra of the symmetric group. We give a conjecture for the case of crepant resolutions of symplectic quotient singularities.
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Arithmétique des schémas en groupes de Bruhat-Tits sur un anneau de Dedekind semi-local
本文为半局部戴德金环上布吕阿-蒂茨群概形的上同调研究奠定基础，给出格罗滕迪克-塞尔猜想的简化证明，并推广到半单单连通群情形。
Sur la cohomologie de la puissance symetrique du fibre tautologique sur le schema de Hilbert ponctuel d'une surface
本文计算了复光滑射影曲面上点希尔伯特格式上重言丛张量行列式丛的对称幂的上同调空间。
Cohomologie et $K$-theorie equivariantes des tours de Bott et des varietes de drapeaux. Application au calcul de Schubert
我们确定了等变上同调的结构和Bott塔的$K$ -理论。通过限制，我们得到了类似的结果Bott-Samelson品种。这一结果使我们能够更精确地描述等变上同调和$K$ -理论的旗品种$X$。给出了$H_T^*(X)$中上积的表达式。
Logarithmic De Rham, Infinitesimal and Betti Cohomologies
本文分析了fs的Log De Rham上同调（不需要对数平滑）$\mathbb C$上的对数方案$Y$（对于$Y$，允许精确闭浸入到$\mathbb C$上的fs log光滑log方案中）、其Log无穷小上同调$H^{^.}(Y^{log}_{inf}, \mathcal O_{Y^{log}_{inf}})$和其Log Betti上同调，其Log Betti上同调是其关联的Kato-Nakayama拓扑空间$Y^{an}_{log}$的上同调，我们证明了它们是同构的。这些结果是两个经典比较定理的对数格式类似。
This book contains notes of a seminar on Ofer Gabber's work on the etale cohomology and uniformization of quasi-excellent schemes. His main results include (cf. introduction) constructibility theorems (for abelian or non-abelian coefficients), vanishing theorems (e.g. affine Lefschetz), uniformization for the "prime-to-l alteration topology", rigidity for non-abelian coefficients, a new proof of the absolute purity conjecture, duality, etc.
Sur la cohomologie des faisceaux l-adiques entiers sur les corps locaux
研究了局部域上有限型格式的积分l-adic层在六种运算和邻近的循环函子下的行为。--
We establish a structure theorem on the arc space of a $k$-scheme of finite type. More precisely, we show that the arc space is locally for the pro-smooth toplogy a product of an infinite dimensional affine space and of a non-noetherian scheme, stratified by $k$-schemes of finite type, that glues the different formal completions of the arc space. This statement globalizes Drinfeld-Grinberg-Kazhdan's theorem and proves a conjecture of Kollar and Nemethi. Then, we use this theorem to construct a derived category of constructible sheaves, stable by six operations and we show a finiteness result f…
Syntomic cohomology here defined yields a link between rigid cohomology and etale cohomology, viewing the last one as the fixed points under Frobenius of the former one. Let V be a complete discrete valuation ring, with perfect residue field k = V/m of characteristic p > 0 and fraction field K of characteristic 0. Having defined syntomic cohomology with compact supports of an abelian sheaf G on a k-scheme X, we show that it coincides with etale cohomology with compact supports when G is a lisse sheaf. If moreover the convergent F-isocrystal associated to G comes from an overconvergent isocryst…
Sur la cohomologie d'un fibre tautologique sur le schema de Hilbert d'une surface
本文计算了光滑射影曲面X的长为n的子概型的点希尔伯特概型H上的重言式丛张量行列式丛的上同调空间。本文证明了对X上的可逆向量丛L和A，以及X的典范丛，如果$w^{-1}\otimes L$，$w^{-1}\otimes A$和A是充分向量丛，则L张量所对应的重言式丛和A所对应的行列式丛在H上的高上同调空间为零，并且整体瓣空间是用$H^0(A)$和$H^0(L\otimes A)$计算的.这个结果是由计算射影平面上秩为2的半稳定层的模空间上的行列式丛的整体瓣空间所得到的，支持了射影平面上的Le Potier的Strange对偶猜想.
We define, for a regular scheme $S$ and a given field of characteristic zero $\KK$, the notion of $\KK$-linear mixed Weil cohomology on smooth $S$-schemes by a simple set of properties, mainly: Nisnevich descent, homotopy invariance, stability (which means that the cohomology of $\GG_{m}$ behaves correctly), and Künneth formula. We prove that any mixed Weil cohomology defined on smooth $S$-schemes induces a symmetric monoidal realization of some suitable triangulated category of motives over $S$ to the derived category of the field $\KK$. This implies a finiteness theorem and a Poincaré dualit…
Theory of weights for log convergent cohomologies II: the case of a proper SNCL scheme in characteristic $p>0$
本文在混合特征情形下，为真简单正规交叉对数概形构造了两个典范同构的滤过复形，由此导出对数收敛上同调层上的权重滤过，并建立了与作者先前等扎里斯基权重滤过复形的比较定理。
Theory of weights for log convergent cohomologies I: the case of a proper smooth scheme with an SNCD in characteristic p>0
本研究针对特征p>0下带相对简单正规交叉除子的对数概形，定义对数收敛上同调的两类基本滤过复形，证明p进纯度与二者典范同构，得到权谱序列，并建立与此前定义的Zariski侧权滤过复形的比较定理。
We study the cohomology of Jacobians and Hilbert schemes of points on reduced and locally planar curves, which are however allowed to be singular and reducible. We show that the cohomologies of all Hilbert schemes of all subcurves are encoded in the cohomologies of the fine compactified Jacobians of connected subcurves, via the perverse Leray filtration.
Sur l'indépendance de l en cohomologie l-adique sur les corps locaux
Gabber从有限域上的一个一般稳定性结果出发，导出了交上同调的$l$的独立性定理。在这篇文章中，我们证明了一个类似的一般结果在局部域。更确切地说，我们引入了一个独立的概念，$l$的系统的复体的$l$ -进层的计划有限型的局部域，有限群下的等变。我们通过Grothendieck的六种操作和附近的循环函子建立了它的稳定性。我们的方法导致一个新的证明的Gabber定理。我们也给出了代数栈的推广。- Gabber提出了$l$的独立性理论，该理论是关于一个总体稳定性结果的交叉点的同调理论。在这篇文章中，我们对一般结果的地方进行了类比。此外，我们还简要地介绍了$l$独立性的概念，以用于$l$ -adiques fini类型的复杂系统，这些复杂系统是由Grothendieck的六个操作和周期过程的函数组成的。我们的方法是获得加伯的新理论。我们也要在阿尔卑斯山上做一个整体。
Cohomologie Automorphe et Compactifications partielles de certaines variétés de Griffiths-Schmid
研究了三元酉群上的Griffiths-Schmid簇的一次自同构上同调类.利用K. Kato和S.因此，我们定义了这样一个上同调类的一些类似的Fourier-Shimura系数，这是上同调类的某些椭圆曲线。我们表明，一个大的空间，这样的自守类可以产生的那些合理的“系数”。更确切地说，我们认为这些上同调类，来自皮卡德模形式，通过一些彭罗斯式的变换研究在以前的文章：我们证明，这样得到的类的系数可以计算从皮卡德形式的系数通过一个类似的变换定义在椭圆曲线的水平。
Let $X$ be a fine and saturated log scheme, and let $G$ be a commutative finite flat group scheme over the underlying scheme of $X$. If $G$-torsors for the fppf topology can be thought of as being unramified objects by nature, then $G$-torsors for the log flat topology allow us to consider tame ramification. Using the results of Kato, we define a concept of Galois structure for these torsors, then we generalize the author's previous constructions (class-invariant homomorphism for semi-stable abelian varieties) in this new setting, thus dropping some restrictions.
We study, using the language of log schemes, the problem of extending biextensions of smooth commutative group schemes by the multiplicative group. This was first considered by Grothendieck in SGA 7. We show that this problem admits a solution in the category of sheaves for Kato's log flat topology, in contradistinction to what can be observed using the fppf topology, for which monodromic obstructions were defined by Grothendieck. In particular, in the case of an abelian variety and its dual, it is possible to extend the Weil biextension to the whole Néron model. This allows us to define a pai…
Cohomology of affinoids does not behave well; often, this can be remedied by making affinoids overconvergent. In this paper, we focus on dimension 1 and compute, using analogs of pants decompositions of Riemann surfaces, various cohomologies of affinoids. To give a meaning to these decompositions we modify slightly the notion of $p$-adic formal scheme, which gives rise to the adoc (an interpolation between adic and ad hoc) geometry. It turns out that cohomology of affinoids (in dimension 1) is not that pathological. From this we deduce a computation of cohomologies of curves without boundary (…
Let $X$ be an algebraic scheme over an algebraically closed field and $\ell$ a prime number invertible on $X$. According to classical results (due essentially to A. Grothendieck, M. Artin and P. Deligne), the étale cohomology groups $\mathrm{H}^i(X,\mathbb{Z}/\ell\mathbb{Z})$ are finite-dimensional. Using an $\ell$-adic variant of M. Artin's good neighborhoods and elementary results on the cohomology of pro-$\ell$ groups, we express the cohomology of $X$ as a well controlled colimit of that of toposes constructed on $BG$ where the $G$ are computable finite $\ell$-groups. From this, we deduce t…
Let $k$ be a perfect field of characteristic $p > 0$, $W_n = W_n(k)$. For separated $k$-schemes of finite type, we explain how rigid cohomology with compact supports can be computed as the cohomology of certain de Rham-Witt complexes with coefficients. This result generalizes the classical comparison theorem of Bloch-Illusie for proper and smooth schemes. In the proof, the key step is an extension of the Bloch-Illusie theorem to the case of cohomologies relative to $W_n$ with coefficients in a crystal that is only supposed to be flat over $W_n$.
This thesis deals with the algorithmic representation of constructible sheaves of abelian groups on the étale site of a variety over an algebraically closed field, as well as the explicit computation of their cohomology. We describe three representations of such sheaves on curves with at worst nodal singularities, as well as algorithms performing various operations (kernels and cokernels of morphisms, pullback and pushforward, internal Hom and tensor product) on these sheaves. We present an algorithm computing the cohomology complex of a locally constant constructible sheaf on a smooth or noda…
$p$-Adic Weight Spectral Sequences of Strictly Semi-stable Schemes over Formal Power Series Rings via Arithmetic $\mathcal{D}$-modules
本文通过算术$\mathcal{D}$-模理论，构建了严格半稳定方案在形式幂级数环上的$p$-adic权重谱序列，其$E_1$项由闭纤维的不可约成分的刚性上同调描述，$E_\infty$项猜想由Lazda-Pál的刚性上同调的unipotent邻近循环描述，并探讨了其泛函性。
Espaces de Berkovich globaux : catégorie, topologie, cohomologie
本文研究整体Berkovich空间的基础理论，定义了性质良好的态射并建立相关构造，证明其局部道路连通性，还证明任意维开/闭圆盘无高阶凝聚上同调，将算术幂级数环的Noether性定理推广到多变量情形。
The Chow rings of generalized Grassmannians
基于Bruhat-Chevalley的基定理和在[DPDMATHABMM]中得到并在[DZ $_{\QTR{group}{1}}$ ]中编程实现的Schubert类的乘法公式，本文提出了一种计算旗簇Chow环的新方法.齐性空间的积分上同调）。本文的方法和结果推广到[DZ $_{3}$，DZ $_{4}$ ]中，得到了完备旗流形的积分上同调环，并构造了李群的Schubert类积分上同调.
For a proper, smooth scheme $X$ over a $p$-adic field $K$, we show that any proper, flat, semistable $\mathcal{O}_K$-model $\mathcal{X}$ of $X$ whose logarithmic de Rham cohomology is torsion free determines the same $\mathcal{O}_K$-lattice inside $H^i_{dR}(X/K)$ and, moreover, that this lattice is functorial in $X$. For this, we extend the results of Bhatt--Morrow--Scholze on the construction and the analysis of an $A_{inf}$-valued cohomology theory of $p$-adic formal, proper, smooth $\mathcal{O}_{\overline{K}}$-schemes $\mathfrak{X}$ to the semistable case. The relation of the $A_{inf}$-coho…
Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface
本文猜想并部分证明了 K3 曲面点的 Hilbert 概形的 Gromov-Witten 不变量生成级数满足全纯反常方程，并揭示了其在超凯勒流形量子上同调及 CHL Calabi-Yau 配分函数中的应用。
We describe the mod $p^r$ pro $K$-groups $\{K_n(A/I^s)/p^r\}_s$ of a regular local $\mathbb F_p$-algebra $A$ modulo powers of a suitable ideal $I$, in terms of logarithmic Hodge-Witt groups, by proving pro analogues of the theorems of Geisser-Levine and Bloch-Kato-Gabber. This is achieved by combining the pro Hochschild-Kostant-Rosenberg theorem in topological cyclic homology with the development of the theory of de Rham-Witt complexes and logarithmic Hodge-Witt sheaves on formal schemes in characteristic $p$. Applications include the following: the infinitesimal part of the weak Lefschetz con…
Crystallinity for syntomic cohomology, étale cohomology, and algebraic $K$-theory
本文证明了模(p^c,v_1^{p^n})的syntomic上同调的晶性，并计算了代数K-理论和etale上同调的结构，同时加强了p-进收敛定理。
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