Varieties In Characteristic Zero Research Articles

Vector bundles on flag varieties

AbstractWe study vector bundles on flag varieties over an algebraically closed field k. In the first part, we suppose to be the Grassmannian parameterizing linear subspaces of dimension d in , where k is an algebraically closed field of characteristic . Let E be a uniform vector bundle over G of rank . We show that E is either a direct sum of line bundles or a twist of the pullback of the universal subbundle or its dual by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles on general flag varieties in characteristic zero are considered. We prove a structure theorem for bundles over flag varieties which are uniform with respect to the ith component of the manifold of lines in . Furthermore, we generalize the Grauert–Mlich–Barth theorem to flag varieties. As a corollary, we show that any strongly uniform i‐semistable bundle over the complete flag variety splits as a direct sum of special line bundles.

Functorial resolution except for toroidal locus. Toroidal compactification

Let X be any variety in characteristic zero. Let V⊂X be an open subset that has toroidal singularities. We show the existence of a canonical desingularization of X except for V. It is a morphism f:Y→X, which does not modify the subset V and transforms X into a toroidal embedding Y, with singularities extending those on V. Moreover, the exceptional divisor has simple normal crossings on Y.The theorem naturally generalizes the Hironaka canonical desingularization. It does not modify the nonsingular locus V and transforms X into a nonsingular variety Y.The proof uses, in particular, the canonical desingularization of logarithmic varieties recently proved by Abramovich-Temkin-Włodarczyk. It also relies on the established here canonical functorial desingularization of locally toric varieties with an unmodified open toroidal subset. As an application, we show the existence of a toroidal equisingular compactification of toroidal varieties. All the results here can be linked to a simple functorial combinatorial desingularization algorithm developed in this paper.

On Separable Higher Gauss Maps

We study the mth Gauss map in the sense of F. L. Zak of a projective variety X⊂PN over an algebraically closed field in any characteristic. For all integers m with n:=dim(X)⩽m<N, we show that the contact locus on X of a general tangent m-plane is a linear variety if the mth Gauss map is separable. We also show that for smooth X with n<N−2, the (n+1)th Gauss map is birational if it is separable, unless X is the Segre embedding P1×Pn⊂P2n−1. This is related to Ein’s classification of varieties with small dual varieties in characteristic zero.

COMPACTIFICATIONS OF SUBSCHEMES OF INTEGRAL MODELS OF SHIMURA VARIETIES

We study several kinds of subschemes of mixed characteristic models of Shimura varieties which admit good (partial) toroidal and minimal compactifications, with familiar boundary stratifications and formal local structures, as if they were Shimura varieties in characteristic zero. We also generalize Koecher’s principle and the relative vanishing of subcanonical extensions for coherent sheaves, and Pink’s and Morel’s formulas for étale sheaves, to the context of such subschemes.

Nearby cycles of automorphic étale sheaves

We show that the automorphic étale cohomology of a (possibly noncompact) PEL-type or Hodge-type Shimura variety in characteristic zero is canonically isomorphic to the cohomology of the associated nearby cycles over most of their mixed characteristics models constructed in the literature.

Gauss maps of toric varieties

We investigate Gauss maps of (not necessarily normal) projective toric varieties over an algebraically closed field of arbitrary characteristic. The main results are as follows: (1) The structure of the Gauss map of a toric variety is described in terms of combinatorics in any characteristic. (2) We give a developability criterion in the toric case. In particular, we show that any toric variety whose Gauss map is degenerate must be the join of some toric varieties in characteristic zero. (3) As applications, we provide two constructions of toric varieties whose Gauss maps have some given data (e.g., fibers, images) in positive characteristic.

Lyubeznik numbers for nonsingular projective varieties

In this paper, we determine completely the Lyubeznik numbers λ i , j ( A ) of the local ring A at the vertex of the affine cone over a nonsingular projective variety V, where V is defined over a field of characteristic zero, in terms of the dimensions of the algebraic de Rham cohomology spaces of V. In particular, we prove that these numbers are intrinsic numerical invariants of V, even though a priori their definition depends on an embedding into projective space. This provides supporting evidence for a positive answer to the question of embedding-independence for arbitrary varieties in characteristic zero, which is still open.

Estimates for F-jumping numbers and bounds for Hartshorne–Speiser–Lyubeznik numbers

AbstractGiven an ideal a on a smooth variety in characteristic zero, we estimate theF-jumping numbers of the reductions of a to positive characteristic in terms of the jumping numbers of a and the characteristic. We apply one of our estimates to bound the Hartshorne–Speiser–Lyubeznik invariant for the reduction to positive characteristic of a hypersurface singularity.

Estimates forF-jumping numbers and bounds for Hartshorne–Speiser–Lyubeznik numbers

AbstractGiven an ideal a on a smooth variety in characteristic zero, we estimate theF-jumping numbers of the reductions of a to positive characteristic in terms of the jumping numbers of a and the characteristic. We apply one of our estimates to bound the Hartshorne–Speiser–Lyubeznik invariant for the reduction to positive characteristic of a hypersurface singularity.

On Additive invariants of actions of additive and multiplicative groups

AbstractLet X be an algebraic variety with an action of either the additive or multiplicative group. We calculate the additive invariants of X in terms of the additive invariants of the fixed point set, using a formula of Białynicki-Birula. The method is also generalized to calculate certain additive invariants for Chow varieties. As applications, we obtain results on the Hodge polynomial of Chow varieties in characteristic zero and the number of points for Chow varieties over finite fields. As applications, we obtain the l-adic Euler-Poincaré characteristic for the Chow varieties of certain projective varieties over a field of arbitrary characteristic. Moreover, we show that the virtual Hodge (p,0) and (0,q)-numbers of the Chow varieties and affine algebraic group varieties are zero for all p,q positive.

Line bundles with partially vanishing cohomology

Define a line bundle L on a projective variety to be q -ample, for a natural number q , if tensoring with high powers of L kills coherent sheaf cohomology above dimension q . Thus 0-ampleness is the usual notion of ampleness. We show that q -ampleness of a line bundle on a projective variety in characteristic zero is equivalent to the vanishing of an explicit finite list of cohomology groups. It follows that q -ampleness is a Zariski open condition, which is not clear from the definition.

A game for the resolution of singularities

We propose a combinatorial game on finite graphs, called Stratify, that is played by two protagonists, 𝒜 and ℬ. The game captures the logical structure of a proof of the resolution of singularities. In each round, the graph of the game is modified by the moves of the players. When it assumes a final configuration, 𝒜 has won. Otherwise, the game goes on forever, and nobody wins. In particular, ℬ cannot win the game, but can only prevent 𝒜 from winning. We show that 𝒜 always possesses a winning strategy, regardless of the initial shape of the graph and of the moves of ℬ. This implies, translating the game back to algebraic geometry, that there is a choice of centres for the blowup of singular varieties in characteristic zero which eventually leads to their resolution. The algebra needed for this implication is elementary. The transcription from varieties to graphs and from blowups to modifications of the graph thus axiomatizes the proof of the resolution of singularities. In principle, the same logic could also work in positive characteristic, once an appropriate descent in dimension is settled.

Ordinary varieties and the comparison between multiplier ideals and test ideals II

We consider the following conjecture: if X is a smooth projective variety over a field of characteristic zero, then there is a dense set of reductions X_s to positive characteristic such that the action of the Frobenius morphism on the top Zariski cohomology of the structure sheaf on X_s is bijective. We also consider the conjecture relating the multiplier ideals of an ideal J on a nonsingular variety in characteristic zero, and the test ideals of the reductions of J to positive characteristic. We prove that the latter conjecture implies the former one. The converse was proved in a joint paper of the author with V. Srinivas.

$Q$-universal desingularization

We prove that the algorithm for desingularization of algebraic varieties in characteristic zero of the first two authors is functorial with respect to regular morphisms. For this purpose, we show that, in characteristic zero, a regular morphism with connected affine source can be factored into a smooth morphism, a ground-field extension and a generic-fibre embedding. Every variety of characteristic zero admits a regular morphism to a Q-variety. The desingularization algorithm is therefore Q-universal or absolute in the sense that it is induced from its restriction to varieties over Q. As a consequence, for example, the algorithm extends functorially to localizations and Henselizations of varieties.

Rigid analytic Picard theorems

We prove a geometric logarithmic derivative lemma for rigid analytic mappings to algebraic varieties in characteristic zero. We use the lemma to give a new and simpler proof (at least in characteristic zero) of Berkovich's little Picard theorem, which says there are no nonconstant rigid analytic maps from the affine line to nonsingular projective curves of positive genus, and of Cherry's result that there are no nonconstant rigid analytic maps from the affine line to Abelian varieties. Furthermore, we use the lemma to prove new theorems of little and big Picard type for dominant mappings, in close analogy with Griffiths and King. For the little Picard type theorem, we prove that if X is a smooth projective variety with a simple normal crossings divisor D such that ( X, D ) has nonnegative logarithmic Kodaira dimension, then there are no dominant rigid analytic maps f from A m to X \ D . For the big Picard type theorem, we prove that if Y is a nonsingular rigid analytic space, E is an effective simple normal crossings divisor on Y , and if X is a smooth projective variety with a simple normal crossings divisor D such that ( X, D ) is of log-general type, then any dominant rigid analytic map f : Y \ E →. X \ D extends to an analytic map from Y to X .

Jet spaces of varieties over differential and difference fields

We prove some new structural results on finite-dimensional differential algebraic varieties and difference algebraic varieties in characteristic zero, using elementary methods involving jet spaces. Some partial results and problems are given in the positive characteristic cases. The impact of these methods and results on proofs of the Mordell-Lang conjecture for function fields will also be discussed.

Zeta Functions and ‘Kontsevich Invariants’ on Singular Varieties

AbstractLet X be a nonsingular algebraic variety in characteristic zero. To an effective divisor on X Kontsevich has associated a certain motivic integral, living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational (smooth, projective) Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the invariant motivic (Igusa) zeta function, associated to a regular function on X, which specializes to both the classical p-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant.This paper treats a generalization to singular varieties. Batyrev already considered such a ‘Kontsevich invariant’ for log terminal varieties (on the level of Hodge polynomials of varieties instead of in the Grothendieck ring), and previously we introduced a motivic zeta function on normal surface germs. Here on any ℚ-Gorenstein variety X we associate a motivic zeta function and a ‘Kontsevich invariant’ to effective ℚ-Cartier divisors on X whose support contains the singular locus of X.

Weak semistable reduction in characteristic 0

Let X->B be a morphism of varieties in characteristic zero. Semistable reduction has been proved for dim(B)=1 (Kempf, Knudsen, Mumford, Saint-Donat), dim(X)=dim(B)-1 (de Jong) and dim(X)=dim(B)+2 (Alexeev, Kollar, Shepherd-Barron). In this paper we consider the general case. First we define what we mean by a semistable morphism in terms of toroidal embeddings. Then we reduce the varieties to toroidal embeddings and solve a slightly weaker version of semistable reduction. We also state the full semistable reduction problem in terms of combinatorics of the associated polyhedral complexes.

Algorithmic computation of local cohomology modules and the local cohomological dimension of algebraic varieties

In this paper we present algorithms that compute certain local cohomology modules associated to ideals in a ring of polynomials containing the rational numbers. In particular, we are able to compute the local cohomological dimension of algebraic varieties in characteristic zero. Our approach is based on the theory of D-modules.