Géométrie diophantienne et variétés toriques

We present some results on projective toric varieties which are relevant in diophantine geometry. We interpret and study several invariants attached to these varieties by geometrical and combinatorial terms. We also give a Bézout theorem for Chow weights of projective varieties and an application to the theorem of successive algebraic minima. These results are extracted from the two texts of Philippon and Sombra mentioned in the references at the end of this Note (both downloadable from http://fr.arxiv.org). To cite this article: P. Philippon, M. Sombra, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On quiver varieties and affine Grassmannians of type A

We construct Nakajima's quiver varieties of type A in terms of affine Grassmannians of type A. This gives a compactification of quiver varieties and a decomposition of affine Grassmannians into a disjoint union of quiver varieties. Consequently, singularities of quiver varieties, nilpotent orbits and affine Grassmannians are the same in type A. The construction also provides a geometric framework for skew (GL(m),GL(n)) duality and identifies the natural basis of weight spaces in Nakajima's construction with the natural basis of multiplicity spaces in tensor products which arises from affine Grassmannians. To cite this article: I. Mirkovi?, M. Vybornov, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Regular principal factors in free objects in varieties in the interval [B 2 ,NB 2 ∨G n ]

The author has shown previously how to associate a completely 0-simple semigroup with a connected bipartite graph containing labelled edges and how to describe the regular principal factors in the free objects in the Rees-Sushkevich varieties RS _n generated by all completely 0-simple semigroups over groups from the Burnside variety G _n of groups of exponent dividing a positive integer n by employing this graphical construction. Here we consider the analogous problem for varieties containing the variety B ₂ , generated by the five element Brandt semigroup B ₂, and contained in the variety NB ₂ ∨G _n where NB ₂ is the variety generated by all left and right zero semigroups together with B ₂. The interval [NB ₂ ,NB ₂ ∨G _n ] is of particular interest as it is an important interval, consisting entirely of varieties generated by completely 0-simple semigroups, in the lattice of subvarieties of RS _n .     

Positive scalar curvature in dim⩾8

We announce a first series of new results and techniques extending the scope of applications of minimal hypersurfaces in scalar curvature geometry. For instance, the restriction to dimensions ?7 which arises from subtle analytic problems in higher dimensions is entirely removed. To cite this article: J. Lohkamp, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

A p-adic proof of Hodge symmetry for threefolds

We give a p-adic proof of Hodge symmetry for smooth and projective varieties of dimension three over the field of complex numbers. To cite this article: K. Joshi, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Classes de Chern des variétés singulières,revisitées

We introduce the notion of a proChow group of varieties, agreeing with the notion of Chow group for complete varieties and covariantly functorial with respect to arbitrary morphisms. We construct a natural transformation from the functor of constructible functions to the proChow functor, extending MacPherson's natural transformation. We illustrate the result by providing very short proofs of (a generalization of) two well-known facts on Chern–Schwartz–MacPherson classes. To cite this article: P. Aluffi, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Diagrams and incidence structures

We show a purity assumption which seems to be implicit in the theory of the geometry of diagrams (developed by F. Buekenhout in: Geom. Dedicata8 (1979), 253–257; 296–298; J. Combin. Theory Ser. A27, (1979), 121–151); characterizations of those structures on which this assumption holds are given (pure structures), and a sufficent condition on a structure to be pure is also presented.     

Chow–Künneth projectors for modular varietiesProjecteurs de Chow–Künneth pour des variétés modulaires

We show the existence of the Chow–Künneth projectors for certain varieties, including Kuga–Shimura varieties of Hilbert modular varieties. The Chow–Künneth projectors of a smooth projective variety are, by definition, mutually orthogonal idempotents of the Chow ring of self-correspondences which give decomposition of the total cohomology of the variety into degree pieces. To cite this article: B.B. Gordon et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 745–750.     

Problems in algebra inspired by universal algebraic geometry

Let Θ be a variety of algebras. In every variety Θ and every algebra H from Θ one can consider algebraic geometry in Θ over H. We also consider a special categorical invariant K _Θ of this geometry. The classical algebraic geometry deals with the variety Θ = Com-P of all associative and commutative algebras over the ground field of constants P. An algebra H in this setting is an extension of the ground field P. Geometry in groups is related to the varieties Grp and Grp-G, where G is a group of constants. The case Grp-F, where F is a free group, is related to Tarski’s problems devoted to logic of a free group. The described general insight on algebraic geometry in different varieties of algebras inspires some new problems in algebra and algebraic geometry. The problems of such kind determine, to a great extent, the content of universal algebraic geometry. For example, a general and natural problem is: When do algebras H ₁ and H ₂ have the same geometry? Or more specifically, what are the conditions on algebras from a given variety Θ that provide the coincidence of their algebraic geometries? We consider two variants of coincidence: 1) K _Θ(H ₁) and K _Θ(H ₂) are isomorphic; 2) these categories are equivalent. This problem is closely connected with the following general algebraic problem. Let Θ⁰ be the category of all algebras W = W(X) free in Θ, where X is finite. Consider the groups of automorphisms Aunt(Θ⁰) for different varieties Θ and also the groups of autoequivalences of Θ⁰. The problem is to describe these groups for different Θ.     

Propriétés génériques des trajectoires singulières

We give genericity results for singular trajectories in sub-Riemannian geometry: generically (in the sense of the Whitney topology), every singular trajectory is of minimal order and of corank 1 and in particular is not of Goh type if the rank of the distribution is greater or equal to 3. We extend these results to control-affine systems. To cite this article: Y. Chitour et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Semistable abelian varieties with good reduction outside 15

We show that there are no non-zero semi-stable abelian varieties over ${{\bf Q}(\sqrt{5})}$ with good reduction outside 3 and we show that the only semi-stable abelian varieties over Q with good reduction outside 15 are, up to isogeny over Q, powers of the Jacobian of the modular curve X ₀(15).     

The André–Oort conjecture for Drinfeld modular varieties

We state an analogue of the André–Oort conjecture for subvarieties of Drinfeld modular varieties, and prove it in two special cases. To cite this article: F. Breuer, C. R. Acad. Sci. Paris, Ser. I 344 (2007).     

Sums of finitely based lattice varieties

This paper is concerned with the question whether the lattice sum (join) V + V′ of two finitely based lattice varieties V and V′ is finitely based. An example is constructed showing that this is not always the case. On the other hand, it is proved that if V ? M and (V′)³ = (N)³, then V + V′ is finitely based. Here M and N are, respectively, the variety of all modular lattices and the variety generated by the pentagon (the five-element nonmodular lattice), and (V)ⁿ is the variety defined by all those identities with n variables or less that hold in V. In particular, M + N, the unique lattice variety that covers M, is finitely based.     

Spaces of Riemannian metrics on open manifolds

We define for the set M of metrics on an open manifold M ⁿ suitable uniform structures, obtain completed spaces ^b,m M or M ^r (I, B _k ), respectively and calculate for each component of M ^r (I, B _k ) the infinitedimensional geometry. In particular, we show that the sectional curvature is non positive.     

Finite generation of lifted p-adic homology with compact supports. Generalization of the weil conjectures to singular,non-complete algebraic varieties

In Section 1, if $\text{O}$ is a c.d.v.r. with quotient field of characteristic zero and residue class field k, if $\text{A}$ is an $\text{O}$-algebra and if $\text{A =}\text{A}$ ?_$\text{O}$k, then for algebraic families X over A that are polynomially properly embeddable over $\text{A}$, we define the lifted p-adic homology with compact supports$\text{H}{}_{\mathrm{h}}{}^{\mathrm{c}}\text{(X,}\text{A}$2 ?_zQ), which are functors with respect to proper maps. In Section 2, it is shown that, if X is an algebraic variety over k (i.e., if A = k), then the lifted p-adic homology of X with compact supports with coefficients in K is finite dimensional over K = quotient field of $\text{O}$. In Section 3, the results of Sections 1 and 2 are used to generalize both the statement and proof of the Weil “Lefschetz Theorem” Conjecture and the statement (but not the proof) of the Weil “Riemann Hypothesis” Conjecture, to non-complete, singular varieties over finite fields. In addition, the Weil zeta function of varieties over finite fields, is generalized by a device which we call the zeta matrices, W^h(X), 0 ≤ h ≤ 2 dim X, of an algebraic variety X, to varieties over even infinite fields of non-zero characteristic. These are used to give formulas for the zeta functions of each variety in an algebraic family, by means of the zeta matrices of an alebraic family. Sketches only are given. In Section 4, some of the material is duplicated, to define a q-adic homology with compact supports, q ≠ characteristic. The definition only makes sense for algebraic varieties; finite generation is proved. And the Weil “Lefschetz Theorem” Conjecture is established, even for singular, non-complete varieties, as well as a generalization of the Weil “Riemann Hypothesis” Conjecture. (However, zeta matrices do not make sense q-adically.)In Section 5, some special results are proved about p-adic homology with compact supports on affines. And the Weil “Riemann Hypothesis” conjecture is proved p-adically, p = characteristic, for projective, non-singular liftable varieties.     

A note on congruences for theta divisors

The classes of two theta divisors on an Abelian variety in the naive Grothendieck ring of varieties need not be congruent modulo the class of the affine line. To cite this article: F. Heinloth, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Une suite spectrale de Hochschild–Serre pour l'uniformisation de Rapoport–ZinkAn Hochschild–Serre spectral sequence for the Rapoport–Zink uniformization

In this Note we announce results concerning the first part of a programme intending to generalize the articles [5,7] and thus construct local Langlands correspondences for groups other than GL_n (for example, quasisplit unitary groups) inside the ? adic cohomology of Rapoport–Zink spaces. The method consists in comparing the cohomology of these local objects with that of global objects: Shimura varieties. For this we generalize the spectral sequences constructed in [5] and [4]. A part of these results is quoted in [6]. To cite this article: L. Fargues, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 739–742.     

Virtual Betti numbers of real algebraic varieties

We show that for all i?0 the i-th mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a virtual Betti numberβ_i defined for all real algebraic varieties, such that if Y is a closed subvariety of X then β_i(X)=β_i(X?Y)+β_i(Y). We show by example that there is no natural weight filtration on the $\text{Z}{}_2$-cohomology of real algebraic varieties with compact supports such that the virtual Betti numbers are the weighted Euler characteristics. To cite this article: C. McCrory, A. Parusiński, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Équations différentielles stochastiques conduites par des lacets dans les groupes de Carnot

The subriemannian geometry of stochastic differential equations driven by processes generating loops in free Carnot groups are studied. To cite this article: F. Baudoin, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Rings of invariants for representations of quivers

In this Note we compute the generators of the ring of invariants for quiver factorization problems, generalizing results of Le Bruyn and Procesi. In particular, we find a necessary and sufficient combinatorial criterion for the projectivity of the associated invariant quotients. Further, we show that the non-projective quotients admit open immersions into projective varieties, which still arise from suitable quiver factorization problems. To cite this article: M. Halic, M.S. Stupariu, C. R. Acad. Sci. Paris, Ser. I 340 (2005).