Masaki Kashiwara, Niels Lauritzen
Comparing vanishing of local cohomology in zero and positive characteristic, we give an example of a D-module on a Grassmann variety in positive characteristic with non-vanishing first cohomology group. This is a counterexample to D-affinity and the Beilinson-Bernstein equivalence for flag manifolds in positive characteristic.
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In this paper we prove a cohomology vanishing result for quadrics of low dimension in positive characteristic. This vanishing is a necessary condition for the D-affinity of these varieties.
Let $G$ be a simple algebraic group of type $B_2$ over an algebraically closed field of odd characteristic. We prove that the flag variety $G/B$ is D-affine. This extends an earlier result of H.H.Andersen and M.Kaneda.
Hochschild cohomology of differential operators in positive characteristic
本文在正特征域的光滑簇上计算了Grothendieck微分算子的Hochschild上同调，并表明其结果由Frobenius作用于结构层上同调的导出逆极限给出。
We prove that, if $A$ is a positively graded, graded commutative, local, finite Hopf algebra, its cohomology is finitely generated, thus unifying classical results of Wilkerson and Hopkins-Smith, and of Friedlander-Suslin. We do this by showing the existence of conormal elementary quotients.
For any two degrees coprime to the rank, we construct a family of ring isomorphisms parameterized by GSp(2g) between the cohomology of the moduli spaces of stable Higgs bundles which preserve the perverse filtrations. As consequences, we prove two structural results concerning the cohomology of Higgs moduli which are predicted by the P=W conjecture in non-abelian Hodge theory: (1) Galois conjugation for character varieties preserves the perverse filtrations for the corresponding Higgs moduli spaces. (2) The restriction of the Hodge-Tate decomposition for a character variety to each piece of th…
Non-abelian cohomology of universal curves in positive characteristic
本文将Hain关于特征零下一般曲线非阿贝尔上同调的结果推广至正特征，计算了泛完备曲线的非阿贝尔上同调，并证明泛n穿孔曲线的etale基本群正合序列不可裂。
The aim of this paper is to extend the main result of C. Huneke and G. Lyubeznik in [Adv. Math. 210 (2007), 498--504] to the class of rings that are images of Cohen-Macaulay local rings. Namely, let $R$ be a local Noetherian domain of positive characteristic that is an image of a Cohen-Macaulay local ring. We prove that all local cohomology of $R$ (below the dimension) maps to zero in a finite extension of the ring. As a direct consequence we obtain that the absolute integral closure of $R$ is a big Cohen-Macaulay algebra. Since every excellent local ring is an image of a Cohen-Macaulay local …
We give a description of certain local cohomology invariants (introduced by Lyubeznik) for a class of mildly singular varieties in positive characteristic. The novelty of our approuch lies in using the new Riemann-Hilbert type correspondenc of Emerton and Kisin to express these invariants in terms of 'etale cohomology with ZZ/pZZ coefficients. Thus we are able to obtain results in positive characteristic whereas earlier approaches were restricted (due to the use of de Rham theory) to the analytic case.
Even-carry polynomials and cohomology of line bundles on the incidence correspondence in positive characteristic
本文研究正特征下 incidence 对应上线丛的上同调，提出偶进位多项式解决 Liu-Gao-Raicu 的递推关系，得到 n=3 时的显式公式，并推测一般情况下的上同调结构。
We show that in positive characteristic special loci of deformation spaces of rank one $\ell$-adic local systems are quasilinear. From this we deduce the Hard Lefschetz theorem for rank one $\ell$-adic local systems and a generic vanishing theorem. Last version: a few typos corrected
Vanishing results for the coherent cohomology of automorphic vector bundles over the Siegel variety in positive characteristic
本文证明了正特征Siegel簇上自守向量丛的凝聚上同调在特定权条件下的消失结果，并提供了Sage算法以显式计算。
Let $D$ be the ring of Grothendieck differential operators of the ring $R$ of polynomials in $d\geq3$ variables with coefficients in a perfect field of positive characteristic $p.$ We compute the $D$-module length of the first local cohomology module $H^1_f(R)$ of $R$ with respect to an irreducible polynomial $f$ with an isolated singularity, for $p$ large enough. The expression we give is in terms of the Frobenius action on the top coherent cohomology of the structure sheaf of the exceptional divisor of a resolution of the singularity. Our proof rests on a tight closure computation due to Har…
Local Tate duality for syntomic cohomology in positive characteristic
本文针对特征p>0的局部域k，证明带系数且基于Dodd的F-规范理论的合成上同调满足局部塔特对偶，还得到其整数环上合成上同调的拉格朗日性质，p>2时可导出相关经典对偶与正交性结果。
$\overline{\partial}$ cohomology of the complement of a semi-positive anticanonical divisor of a compact surface
本文研究具有半正曲率反正则线丛的紧复曲面上，其反正则除子补集的$\overline{\partial}$上同调群$H^1(M,\mathcal{O}_M)$的结构与性质。
We show that the set of totally positive unipotent lower-triangular Toeplitz matrices in $GL_n$ form a real semi-algebraic cell of dimension $n-1$. Furthermore we prove a natural cell decomposition for its closure. The proof uses properties of the quantum cohomology rings of the partial flag varieties of $GL_n(\C)$ relying in particular on the positivity of the structure constants, which are enumerative Gromov--Witten invariants. We also give a characterization of total positivity for Toeplitz matrices in terms of the (quantum) Schubert classes. This work builds on some results of Dale Peterso…
Let X be a noetherian scheme defined over an algebraically closed field of positive characteristic p, and G be a finite group, of order divisible by p, acting on X. We introduce a refinement of the equivariant K-theory of X to take into account the information related to modular representation theory. As an application, in the 1-dimensional case, we generalize a modular Riemann-Roch theorem given by S.Nakajima, extending the link between Galois modules and wild ramification.
Some applications of the Nygaard filtration and quasisyntomic descent in positive characteristic
本文阐述了正特征下的quasisyntomic下降与Nygaard滤过，提出了Illusie比较定理的新方法，重新审视了Ogus比较定理，并确定了乘以$n$在Abel簇fppf上同调上的作用。
On The Cohomology of $N_C(-2)$ in Positive Characteristic
本文将一般 Brill--Noether 曲线二次截面控制条件 $H^0(N_C(-2))=0$ 的经典结果从特征零推广到正特征，并发现特征 2 下存在大量新例外。
Non-vanishing of the structure-sheaf cohomology of affine henselian schemes and henselian affinoids
本文证明特征零 Hensel 对上的平坦仿射 Hensel 概形在纤维正维时第一结构层上同调非零，否定 Devadas 混合特征问题，并推广至 Hensel 仿射体。
Peterson varieties are a special class of Hessenberg varieties that have been extensively studied e.g. by Peterson, Kostant, and Rietsch, in connection with the quantum cohomology of the flag variety. In this manuscript, we develop a generalized Schubert calculus, and in particular a positive Chevalley-Monk formula, for the ordinary and Borel-equivariant cohomology of the Peterson variety $Y$ in type $A_{n-1}$, with respect to a natural $S^1$-action arising from the standard action of the maximal torus on flag varieties. As far as we know, this is the first example of positive Schubert calculu…
Infinitesimal deformations of opers in positive characteristic and the de Rham cohomology of symmetric products
本文在正特征情形建立了 opers 的 Eichler-Shimura 型分解，证明由 SL₂-oper 诱导的 G-oper 形变空间可分解为其对称积的抛物 de Rham 上同调群直和，并推广到休眠 opers。
In this paper, we study the properties of noetherian rings with uniform annihilators. It turns out that all these rings should be universally catenary and locally equidimensional. We will give a necessary and sufficient condition for these rings, which enable us to prove that if a ring R is the image of a Cohen-Macaulay ring of finite dimension, then R has a uniform annihilator of local cohomology. Moreover, we will give a positive answer to the conjecture [Hu, Conjecture 2.13] about the excellent rings with dimension no more than 5.
Symmetry in Equivariant cohomology of $\mathbb{P}^n$
本文通过提供对称且正的产品规则，解决Anderson和Fulton提出的问题，探讨了投射空间等变上同调的对称性质。
Integral p-adic cohomology theories for open and singular varieties
本文研究正特征下开簇与奇异簇的整 p 进上同调，分别借助 cdp/cdh 拓扑及分裂正常一般 étale 超覆盖，给出次数 1 的良好理论并说明其不能推广到高次。
We start with a curve over an algebraically closed ground field of positive characteristic $p>0$. By using specialization techniques, under suitable natural coprimality conditions, we prove a cohomological Simpson Correspondence between the moduli space of Higgs bundles and the one of connections on the curve. We also prove a new $p$-multiplicative periodicity concerning the cohomology rings of Dolbeault moduli spaces of degrees differing by a factor of $p$. By coupling this $p$-periodicity in characteristic $p$ with lifting/specialization techniques in mixed characteristic, we find, in arbitr…
We describe recent work on positive descriptions of the structure constants of the cohomology of homogeneous spaces such as the Grassmannian, by degenerations and related methods. We give various extensions of these rules, some new and conjectural, to K-theory, equivariant cohomology, equivariant K-theory, and quantum cohomology.
We determine the mod p cohomological invariants for several affine group schemes G in chararacteristic p. These are invariants of G-torsors with values in etale motivic cohomology, or equivalently in Kato's version of Galois cohomology based on differential forms. In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2. We also determine all operations on the mod p etale motivic cohomology of fields, extending Vial's computations of the operations on the mod p Milnor K-theory …
We show that a cone theorem for ${\mathbbA}^1-homotopy invariant contravariant functors implies the vanishing of the positive degree part of the operational Chow cohomology rings of a large class of affine varieties. We also discuss how this vanishing relates to a number of questions about representing Chow cohomology classes of GIT quotients in terms of equivariant cycles.
This is a survey of some recent developments concerning the p-adic cohomology of algebraic varieties over fields of positive characteristic and local fields of mixed characteristic, plus some related areas like p-adic Hodge theory.
Non-liftable varieties via etale cohomology rings
研究在正特征下构造光滑射影簇，通过其\(\mathbb{Q}_{\ell}\)系数平展上同调环性质，给出零特征可提升性新阻碍例子。
We give a proof of a result of D. Peterson's identifying the quantum cohomology ring of a Grassmannian with the reduced coordinate ring of a certain subvariety of $GL_n$. The totally positive part of this subvariety is then constructed and we give closed formulas for the values of the Schubert basis elements on the totally positive points. We then use the developed methods to give a new proof of a formula of Vafa and Intriligator and Bertram for the structure constants (Gromov--Witten invariants). Finally, we use the positivity of these Gromov--Witten invariants to prove certain inequalities f…
Birational and $\mathbf{A}^1$-invariant lattices in the cohomology of the structure sheaf over non-archimedean fields
本文研究了非阿基米德域上光滑和正则方案的结构层上同调，并将其细化为光滑（不一定正则）方案上的$\mathbf{A}^1$-不变上同调理论，同时在正特征情况下，维度不超过3时也成立。作为应用之一，得到了正则光滑变种的函数域的自同构群在结构层上同调上的准单性作用。
We generalize our puzzle formula for ordinary Schubert calculus on Grassmannians, to a formula for the T-equivariant Schubert calculus. The structure constants to be calculated are polynomials in {y_{i+1} - y_i}; they were shown (abstractly) to have positive coefficients in [Graham] math.AG/9908172. Our formula is the first to be manifestly positive in this sense. In particular this gives a new and self-contained proof of the ordinary puzzle formula, by an induction backwards from the "most equivariant" case. The proof of the formula is mostly combinatorial, but requires no prior combinatorics…
We give positive formulas for the restriction of a Schubert Class to a T-fixed point in the equivariant K-theory and equivariant cohomology of the Lagrangian Grassmannian. Our formulas rely on a result of Ghorpade-Raghavan, which gives an equivariant Grobner degeneration of a Schubert variety in the neighborhood of a T-fixed point of the Lagrangian Grassmannian.
This paper is a continuation of the authors article "Enlargements of schemes" (Log. Anal.1 (2007), no. 1, 1-60) We mainly study the behaviour of etale cohomology, algebraic cycles and motives under ultraproducts respectively enlargements. The main motivation for that is to find methods to transfer statements about etale cohomology and algebraic cycles from characteristic zero to positive characteristic and vice versa. We give one application to the independence of $l$ of Betti numbers in etale cohomology and applications to the complexity of algebraic cycles.
Equidistribution towards the Green current in big cohomology classes
本文研究了紧致凯勒流形上亚纯自映射的光滑(1,1)形式和正闭电流的拉回序列的极限分布，探讨了其在大上同调类中的等分布性质。
Demailly showed that the Hodge conjecture is equivalent to the statement that any (p,p)-dimensional closed current with rational cohomology class can be approximated by linear combinations of integration currents associated to subvarieties, and asked whether any strongly positive (p,p)-dimensional closed current with rational cohomology class can be approximated by positive linear combinations of integration currents associated to subvarieties. Using tropical geometry, we construct a (p,p)-dimensional current on a smooth projective variety that does not satisfy the latter statement.
We give positive formulas for the restriction of a Schubert Class to a T-fixed point in the equivariant K-theory and equivariant cohomology of the Grassmannian. Our formulas rely on a result of Kodiyalam-Raghavan and Kreiman-Lakshmibai, which gives an equivariant Grobner degeneration of a Schubert variety in the neighborhood of a T-fixed point of the Grassmannian.
Let $V$ be a complete discrete valuation ring with residue field $k$ of positive characteristic and with fraction field $K$ of characteristic 0. We clarify the analysis behind the Monsky--Washnitzer completion of a commutative $V$-algebra using completions of bornological $V$-algebras. This leads us to a functorial chain complex for a finitely generated commutative algebra over the residue field $k$ that computes its rigid cohomology in the sense of Berthelot.
Let S be an essentially smooth scheme over a field and l a prime number invertible on S. We show that the algebra of bistable operations in the mod l motivic cohomology of smooth S-schemes is generated by the motivic Steenrod operations. This was previously proved by Voevodsky for S a field of characteristic zero. We follow Voevodsky's proof but remove its dependence on characteristic zero by using étale cohomology instead of topological realization and by replacing resolution of singularities with a theorem of Gabber on alterations.
Following an idea of Totaro, we prove that the classical integral cycle class map from algebraic cycles to étale cohomology factors through a quotient of $\ell$-adic étale cobordism over an algebraically closed field of positive characteristic. This shows that there is a strong topological obstruction for cohomology classes to be algebraic and that examples of Atiyah, Hirzebruch and Totaro also work in positive characteristic.
In this paper, we prove a general principle of lifting an automorphism from positive characteristic to zero characteristic. We based on the principle to prove the automorphism group of Fano variety of cubic threefold (fourfold) acts on its cohomology group faithfully. We also prove the same result for a singular cubic threefold (fourfold) with one ordinary double singularity. As an application, we classify the automorphism groups of cubic fourfolds with small middle Picard number over complex numbers.
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