Donu Arapura
When X is a singular complex projective variety, we prove a Kodaira type vanishing theorem generalizing results of Navarro Aznar and others. This is proved by extending the Deligne-Illusie decomposition to the Du Bois complex. We also give a new definition of Frobenius amplitude (introduced in math.AG/0202129) using ultraproducts.
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We prove several asymptotic vanishing theorems for Frobenius twists of ample vector bundles in positive characteristic. As an application, we prove a generalization of the Bott-Danilov-Steenbrink vanishing theorem for ample vector bundles on toric varieties.
Given a smooth formal scheme over the ring of integers of a mixed-characteristic perfectoid field, we study its $p$-adic vanishing cycles via de Rham--Witt and $q$-de Rham complexes.
This paper is about sheaf cohomology for varieties (schemes) in characteristic $p>0$. We assume the presence of a Frobenius splitting. (See V.B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Annals of Math. 122 (1985), 27--40). The main result is that a non-zero higher direct image under a proper map of the ideal sheaf of a compatibly Frobenius split subvariety can not have a support whose inverse image is contained in that subvariety. Earlier vanishing theorems for Frobenius split varieties were based on direct limits and Serre's vanishing theor…
Let $X$ be a projective scheme over a field. We show that the vanishing cohomology of any sequence of coherent sheaves is closely related to vanishing under pullbacks by the Frobenius morphism. We also compare various definitions of ample locally free sheaf and show that the vanishing given by the Frobenius morphism is, in a certain sense, the strongest possible. Our work can be viewed as various generalizations of the Serre Vanishing Theorem.
The Frobenius--stable version of the Grauert--Riemenschneider vanishing theorem fails
本文证明Grauert–Riemenschneider消失定理的Frobenius稳定版本在任意正特征的三fold以及特征p∈{2,3,5}的terminal 3-fold上失效，并引入F_p-理性概念，证明至多4维的klt奇点是F_p-理性的，进而应用于K-平凡3-fold上的Kawamata–Viehweg消失定理的Frobenius稳定版本。
Let $X$ be a smooth projective algebraic variety over $Z/p$, which has a flat lift to a scheme $X'$ over $Z/p^2$. If the absolute Frobenius morphism $F$ on $X$ lifts to a morphism on $X'$, then an old trick by Mazur shows that push-down of the de Rham complex under $F$ decomposes. We show that the quasi-isomorphism in question is split. This is then applied to toric varieties (where a glueing argument gives lifting of Frobenius to $Z/p^2$) and we derive natural characteristic $p$ proofs of Bott vanishing and degeneration of the Danilov spectral sequence. For flag varieties we obtain generaliza…
The primary goal of this paper is to systematically exploit the method of Deligne-Illusie to obtain Kodaira type vanishing theorems for vector bundles and more generally coherent sheaves on algebraic varieties. The key idea is to introduce a number which provides a cohomological measure of the positivity of a coherent sheaf called the Frobenius or F-amplitude. As the terminology indicates, the definition is most natural over a field of positive characteristic, however it can be forced into characteristic 0 by standard tricks. The F-amplitude enters into the statement of the basic vanishing the…
Mumford's well-known characterization of the hyperelliptic locus of the moduli space of ppavs in terms of vanishing and non-vanishing theta constants is based on Neumann's dynamical system. Poor's approach to the characterization uses the cross ratio. A key tool in both methods is Frobenius' theta formula, which follows from Riemann's theta formula. In a 2004 paper Grushevsky gives a different characterization in terms of cubic equations in second order theta functions. In this note we first show the connection between the methods by proving that Grushevsky's cubic equations are strictly relat…
Lakshmibai, Mehta and Parameswaran (LMP) introduced the notion of maximal multiplicity vanishing in Frobenius splitting. In this paper we define the algebraic analogue of this concept and construct a Frobenius splitting vanishing with maximal multiplicity on the diagonal of the full flag variety. Our splitting induces a diagonal Frobenius splitting of maximal multiplicity for a special class of smooth Schubert varieties first considered by Kempf. Consequences are Frobenius splitting of tangent bundles, of blow-ups along the diagonal in flag varieties along with the LMP and Wahl conjectures in …
We prove a generic vanishing type statement in positive characteristic and apply it to prove positive characteristic versions of Kawamata's theorems: a characterization of smooth varieties birational to ordinary abelian varieties and the surjectivity of the Albanese map when the Frobenius stable Kodaira dimension is zero.
We give elementary counterexamples to Kodaira vanishing in prime characteristic constructed as the projectivization of Frobenius pull-backs of natural bundles on the incidence variety of points and planes in projective n-space (n>2).
For a proper (not necessarily smooth) variety over a finite field with q elements, Berthelot-Bloch-Esnault proved a trace formula which computes the number of rational points modulo q in terms of the Witt vector cohomology. We show the analogous formula for Witt vector cohomology of finite length. In addition, we prove a vanishing result for the compactly supported étale cohomology of a constant p-torsion sheaf on an affine Cohen-Macaulay variety in positive characteristic p.
Frobenius--Witt cotangent complex for derived rings
本文对Frobenius--Witt余切复形的算术扩张给出拉回描述，刻画了该复形并将其推广到导出环等，还建立了棱柱的消失结果，推广了完美畴环的相关消失性。
We use the liftability of the relative Frobenius morphism of toric varieties and the strong liftability of toric varieties to prove the Bott vanishing theorem, the degeneration of the Hodge to de Rham spectral sequence and the Kawamata-Viehweg vanishing theorem for log pairs on toric varieties in positive characteristic. These results generalize those results of Danilov, Buch-Thomsen-Lauritzen-Mehta, Mustata and Fujino to the case where concerned Weil divisors are not necessarily torus invariant.
A Frobenius splitting and cohomology vanishing for the cotangent bundles of the flag varieties of GL_n
本文证明了 GL_n 旗丛余切丛的 Frobenius 分裂具有最高次数，给出了分量函数公式并证明其在例外轨迹上消失，进而得出与支配权相关线丛的高阶上同调群为零。
Bogomolov type inequalities and Frobenius semipositivity
本文证明了满足 Frobenius 半正性的半稳定层的高阶 Chern 特征的 Bogomolov 型不等式，并应用于射影空间和阿贝尔簇中小余维数簇的 Chern 特征不等式。
An Ogus Principle for Zip period maps: The Hasse invariant's vanishing order via `Frobenius and the Hodge filtration'
本文将 Ogus 关于 Calabi–Yau 族 Hasse 不变量消失阶与共轭线位置的结果推广到群论 Zip 周期映射，并导出 Shimura 簇及 K3 模空间上的原理。
Frobenius pushforwards of of vector bundles on projective spaces
本文研究 Beilinson 谱序列诱导滤过的非典范分裂，证明了高阶 Frobenius 前推可分解为线丛与余切丛外幂的直和，并简证了 Klyachko 定理。
We use the G-invariant non-degenerate form on the Steinberg module to Frobenius split the cotangent bundle of a flag variety in good prime characteristics. This was previously only known for the general linear group. Applications are a vanishing theorem for pull back of line bundles to the cotangent bundle (proved for the classical groups and G_2 by Andersen and Jantzen and in characteristic zero by B. Broer (for all groups)), normality and rational singularities for the subregular nilpotent variety and good filtrations of the global sections of pull backs of line bundles to the cotangent bund…
Frobenius type positivity of Hodge bundles and applications
本文为相干层定义F^t-丰沛性和F^tGG-丰沛性两种Frobenius型正性概念，证明它们强于通常丰沛性，并建立半稳定态射下Hodge丛的F^1GG-半正性，推广了Fujita等人的结果，得到顶Chern特征正性及消失定理，并应用于Fano簇像和斜率不等式研究。
Let π: V \rightarrow M be a (real or holomorphic) vector bundle whose base has an almost Frobenius structure (\circ_{M},e_{M}, g_{M}) and typical fiber has the structure of a Frobenius algebra (\circ_{V},e_{V},g_{V}). Using a connection D on the bundle V and a morphism α: V \rightarrow TM, we construct an almost Frobenius structure (\circ,e_{V},g) on the manifold V and we study when it is Frobenius. We describe all (real) positive-definite Frobenius structures on V, obtained in this way, when M is a semisimple Frobenius manifold with non-vanishing rotation coefficients. In the holomorphic sett…
Let $X$ be a smooth variety over an algebraically closed field $k$ of positive characteristic, ${\rm D}_X$ the sheaf of PD-differential operators, and ${\bar D}_X$ its central reduction, the sheaf of small differential operators. In this paper we show that if $X$ is a line-hyperplane incidence variety (a partial flag variety of type $(1,n,n+1)$) or a quadric of arbitrary dimension (in this case the characteristic is supposed to be odd) then ${\rm H}^{i}(X,{\bar D}_X)=0$ for $i>0$. Using this vanishing result and the derived localization theorem for crystalline differential operators (\cite{BMR…
Uniform stratified vanishing and equidistribution on $\mathbb{G}_m^d$
该研究针对有限域上$\boldsymbol{\text{G}}_m^d$的反常层，证明一致分层一般消失定理并推广Katz的等分布定理，得到Frobenius共轭类在极大紧子群共轭类空间中等分布的结论。
This is a sequel to the paper "Frobenius amplitude and strong vanishing theorems for vector bundles" (math.AG/0202129). We introduce a more elementary variant of the notion of F-amplitude from the earlier paper which we call amplitude. This provides another measure of positivity of a vector bundle which is related to a number of preexisting positivity notions such as k-ampleness or q-convexity. We use this to refine the estimates of F-amplitude from the first paper, and to deduce some further vanishing theorems as a consequence. We also give some new proofs of some known vanishing theorems for…
We define weighted projective Reed-Muller codes over a subset of weighted projective space over a finite field. We focus on the case when the set X is a projective weighted torus. We show that the vanishing ideal of X is a lattice ideal and relate it with the lattice ideal of a minimal presentation of the semigroup algebra of Q, the numerical semigroup generated by the weights of the projective space. We compute the index of regularity of the vanishing ideal as function of the weights and the Frobenius number of Q. We compute the basic parameters of weighted projective Reed-Muller codes over a…
We show that the Frobenius character of the equivariant Borel-Moore homology of a certain positive $GL_n$-version of the unramified affine Springer fiber $Z_k$ studied by Goreski, Kottwitz and MacPherson is computed by the matrix coefficients of the $\nabla^k$-operator, which acts diagonally in the modified Macdonald basis. We do this by relating the combinatorial formula for the $\nabla^k$-operator we obtained in an earlier paper to the GKM paving of $Z_k$, and we give an algebraic presentation of the above homology as an explicit submodule of the Kostant-Kumar nil Hecke algebra. We then stud…
For a finite dimensional vector space V of dimension n, we consider the incidence correspondence (or partial flag variety) X in P(V) x P(V*), parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on X in characteristic p>0. If n=3 then X is the full flag variety of V, and the characterization is contained in the thesis of Griffith from the 70s. In characteristic 0, the cohomology groups are described for all V by the Borel-Weil-Bott theorem. Our strategy is to recast…
Bornes de torsion et un théorème effectif du pgcd
本文证明了 Deligne 的“最大公约数定理”的一个有效概率版本，适用于有限域上的光滑射影几何整簇，其核心方法是通过界定 Betti 上同调的挠率、利用大单值性结果及 Frobenius 均匀分布来实现。
Non-decomposability of the de Rham complex and non-semisimplicity of the Sen operator
本文研究了特征 $p$ 下 de Rham 复形的分解障碍，证明了存在 Hodge-to-de Rham 谱序列不退化的可提升光滑射影簇，并基于交换代数与 Steenrod 运算揭示了 Sen 算子的非半单性及 $E_{\infty}$-代数形式性条件。
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