Champs affines

The purpose of this work is to introduce a notion of affine stacks, which is a homotopy version of the notion of affine schemes, and to give several applications in the context of algebraic topology and algebraic geometry. As a first application we show how affine stacks can be used in order to give a new point of view (and new proofs) on rational and p-adic homotopy theory. This gives a first solution to A. Grothendieck’s schematization problem described in [18]. We also use affine stacks in order to introduce a notion of schematic homotopy types. We show that schematic homotopy types give a second solution to the schematization problem, which also allows us to go beyond rational and p-adic homotopy theory for spaces with arbitrary fundamental groups. The notion of schematic homotopy types is also used in order to construct various homotopy types of algebraic varieties corresponding to various co-homology theories (Betti, de Rham, l-adic, ...), extending the well known constructions of the various fundamental groups. Finally, just as algebraic stacks are obtained by gluing affine schemes we define $$ \infty $$-geometric stacks as a certain gluing of affine stacks. Examples of $$ \infty $$-geometric stacks in the context of algebraic topology (moduli spaces of dga structures up to quasi-isomorphisms) and Hodge theory (non-abelian periods) are given.     

Paving Hessenberg varieties by affines

Regular nilpotent Hessenberg varieties form a family of subvarieties of the flag variety arising in the study of quantum cohomology, geometric representation theory, and numerical analysis. In this paper we construct a paving by affines of regular nilpotent Hessenberg varieties for all classical types, generalizing results of De Concini–Lusztig–Procesi and Kostant. This paving is in fact the intersection of a particular Bruhat decomposition with the Hessenberg variety. The nonempty cells of the paving and their dimensions are identified by combinatorial conditions on roots. We use the paving to prove these Hessenberg varieties have no odd-dimensional homology.      

Multiplication maps and vanishing theorems for Toric varieties

We use multiplication maps to give a characteristic-free approach to vanishing theorems on toric varieties. Our approach is very elementary but is enough powerful to prove vanishing theorems.     

Cohomology of the moduli space of Hecke cycles

Let X be a smooth projective curve of genus g?3 and let M₀ be the moduli space of semistable bundles over X of rank 2 with trivial determinant. Three different desingularizations of M₀ have been constructed by Seshadri (Proceedings of the International Symposium on Algebraic Geometry, 1978, 155), Narasimhan-Ramanan (C. P. Ramanujam—A Tribute, 1978, 231), and Kirwan (Proc. London Math. Soc. 65(3) (1992) 474). In this paper, we construct a birational morphism from Kirwan's desingularization to Narasimhan-Ramanan's, and prove that the Narasimhan-Ramanan's desingularization (called the moduli space of Hecke cycles) is the intermediate variety between Kirwan's and Seshadri's as was conjectured recently in (Math. Ann. 330 (2004) 491). As a by-product, we compute the cohomology of the moduli space of Hecke cycles.     

Towards the cohomology of moduli spaces of higher genus stable maps

We prove that the orbifold desingularization of the moduli space of stable maps of genus g = 1 recently constructed by Vakil and Zinger has vanishing rational cohomology groups in odd degree k < 11. Received: 29 January 2007     

Cohomology of toric bundles

Let p: E B be a principal bundle with fibre and structure group the torus T ( ^*)ⁿ over a topological space B. Let X be a nonsingular projective T-toric variety. One has the X-bundle : E(X) B where E(X) = E × _T X, ([e,x]) = p(e). This is a Zariski locally trivial fibre bundle in case p: E B is algebraic. The purpose of this note is to describe (i) the singular cohomology ring of E(X) as an H ^* (B;)-algebra, (ii) the topological K-ring of K ^* (E(X)) as a K ^* (B)-algebra when B is compact. When p : E B is algebraic over an irreducible, nonsingular, noetherian scheme over , we describe (iii) the Chow ring of A ^* (E(X)) as an A ^* (B)-algebra, and (iv) the Grothendieck ring $\mathcal K$⁰ (E (X)) of algebraic vector bundles on E (X) as a $\mathcal K$⁰(B)-algebra.     

A polar de Rham theorem

We prove an analogue of the de Rham theorem for polar homology; that the polar homology HP_q(X) of a smooth projective variety X is isomorphic to its H^n,n−q Dolbeault cohomology group. This analogue can be regarded as a geometric complexification where arbitrary (sub)manifolds are replaced by complex (sub)manifolds and de Rham's operator d is replaced by Dolbeault's .     

Generic character sheaves on reductive groups over a finite ring

In this paper we propose a construction of generic character sheaves on reductive groups over finite local rings at even levels, whose characteristic functions are higher Deligne–Lusztig characters when the parameters are generic. We formulate a conjecture on the simple perversity of these complexes, and we prove it in the level two case (thus extend a result of Lusztig from the function field case). We then discuss the induction and restriction functors, as well as the Frobenius reciprocity, based on the perversity.     

Stanley-Reisner rings and the radicals of lattice ideals

In this article we associate to every lattice ideal I_L,ρ⊂K[x₁,…,x_m] a cone σ and a simplicial complex Δ_σ with vertices the minimal generators of the Stanley-Reisner ideal of σ. We assign a simplicial subcomplex Δ_σ(F) of Δ_σ to every polynomial F. If F₁,…,F_s generate I_L,ρ or they generate rad(I_L,ρ) up to radical, then is a spanning subcomplex of Δ_σ. This result provides a lower bound for the minimal number of generators of I_L,ρ which improves the generalized Krull's principal ideal theorem for lattice ideals. But mainly it provides lower bounds for the binomial arithmetical rank and the A-homogeneous arithmetical rank of a lattice ideal. Finally, we show by a family of examples that the given bounds are sharp.     

Poincaré automorphisms for nondegenerate CR quadrics

Research was supported by Max-Planck-Institut Bonn     

The spherical spectrum of a graded ring

We introduce a functor Sph, the spherical spectrum, which assigns to a graded ringG a space Sph(G) of homogeneous orderings ofG. It combines ideas of concrete geometry in theN-sphere defined by positively homogeneous polynomial equations and inequalities with the abstract notion of the real spectrum of a ring to give a counterpart for real semialgebraic geometry of the functor Proj.     

Existence de F-cristaux avec structures supplémentaires

In this article, we prove, under some hypothesis of non ramification, a conjecture of Kottwitz and Rapoport giving the existence of crystals with additional structures.     

Relaxation oscillations in a class of predator-prey systems

We consider a class of three-dimensional, singularly perturbed predator-prey systems having two predators competing exploitatively for the same prey in a constant environment. By using dynamical systems techniques and the geometric singular perturbation theory, we give precise conditions which guarantee the existence of stable relaxation oscillations for systems within the class. Such result shows the coexistence of the predators and the prey with quite diversified time response which typically happens when the prey population grows much faster than those of predators. As an application, a well-known model will be discussed in detail by showing the existence of stable relaxation oscillations for a wide range of parameters values of the model.     

On toric varieties which are almost set-theoretic complete intersections

We describe a class of affine toric varieties V that are set-theoretically minimally defined by binomial equations over fields of any characteristic.     

Effectivity of Brauer-Manin obstructions

We study Brauer-Manin obstructions to the Hasse principle and to weak approximation, with special regard to effectivity questions.     

Valuations, equimultiplicity and normal flatness

Let R be a regular noetherian local ring of dimension n≥2 and (R_i)≡R=R₀⊂R₁⊂R₂⊂?⊂R_i⊂? be a sequence of successive quadratic transforms along a regular prime ideal p of R (i.e if p_i is the strict transform of p in R_i, then p_i≠R_i, i≥0). We say that p is maximal for (R_i) if for every non-negative integer j≥0 and for every prime ideal q_j of R_j such that (R_i) is a quadratic sequence along q_j with p_j⊂q_j, we have p_j=q_j. We show that p is maximal for (R_i) if and only if V=∪_i≥0R_i/p_i is a valuation ring of dimension one. In this case, the equimultiple locus at p is the set of elements of the maximal ideal of R for which the multiplicity is stable along the sequence (R_i), provided that the series of real numbers given by the multiplicity sequence associated with V diverges. Furthermore, if we consider an ideal J of R, we also show that is normally flat along at the closed point if and only if the Hironaka’s character ν^∗(J,R) is stable along the sequence (R_i). This generalizes well known results for the case where p has height one (see [B.M. Bennett, On the characteristic functions of a local ring, Ann. of Math. Second Series 91 (1) (1970) 25-87]).