New Beauville surfaces and finite simple groups

In this paper we construct new Beauville surfaces with group either PSL(2, p ^e ), or belonging to some other families of finite simple groups of Lie type of low Lie rank, or an alternating group, or a symmetric group, proving a conjecture of Bauer, Catanese and Grunewald. The proofs rely on probabilistic group theoretical results of Liebeck and Shalev, on classical results of Macbeath and on recent results of Marion.     

Remarks on Serre’s modularity conjecture

In this article we give a proof of Serre’s conjecture for the case of odd level and arbitrary weight. Our proof does not use any modularity lifting theorem in characteristic 2 (moreover, we will not consider at all characteristic 2 representations at any step of our proof). The key tool in the proof is a very general modularity lifting result of Kisin, which is combined with the methods and results of previous articles on Serre’s conjecture by Khare, Wintenberger, and the author, and modularity results of Schoof for abelian varieties of small conductor. Assuming GRH, infinitely many cases of even level will also be proved.     

Special subvarieties of EPW sextics

We study the geometry of EPW sextics in order to produce special subvarieties. In particular we exhibit a (singular) Enriques surface and we compute its class in the Chow ring of the sextic. In order to do this, we produce an explicit degeneration of double EPW sextics to a Hilbert scheme of two points on a quartic surface, both in the smooth and in the singular case (keeping the singularities). The fixed locus of the covering involution on the double EPW sextic degenerates to the surface of bitangents to the quartic, which can be shown to be birational to an Enriques surface, provided the quartic acquires enough nodes. This construction is used in Ferretti (Algebra Number Theory, 2009a) as a starting point to prove a conjecture of Beauville and Voisin on the Chow ring of irreducible symplectic varieties, in the particular case of a very general double EPW sextic.     

The Mukai conjecture for log Fano manifolds

For a log Fano manifold (X,D) with D ≠ 0 and of the log Fano pseudoindex ≥2, we prove that the restriction homomorphism Pic(X) → Pic(D ₁) of Picard groups is injective for any irreducible component D ₁ ? D. The strategy of our proof is to run a certain minimal model program and is similar to Casagrande’s argument. As a corollary, we prove that the Mukai conjecture (resp. the generalized Mukai conjecture) implies the log Mukai conjecture (resp. the log generalized Mukai conjecture).     

Algebraic cycles on Prym varieties

This article proposes a generalization of tautological rings introduced by Beauville and Moonen for Jacobians. The main result is that, under certain hypotheses, the special subvarieties of Prym varieties are algebraically equivalent and their classes belong to the tautological ring.     

The Tate conjecture for K3 surfaces over finite fields

Artin’s conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin’s conjecture over fields of characteristic p≥5. This implies Tate’s conjecture for K3 surfaces over finite fields of characteristic p≥5. Our results also yield the Tate conjecture for divisors on certain holomorphic symplectic varieties over finite fields, with some restrictions on the characteristic. As a consequence, we prove the Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p≥5.     

Small codimension subvarieties in homogeneous spaces

We prove Bertini type theorems for the inverse image, under a proper morphism, of any Schubert variety in an homogeneous space. Using generalisations of Deligne's trick, we deduce connectedness results for the inverse image of the diagonal in X² where X is any isotropic grassmannian. We also deduce simple connectedness properties for subvarieties of X. Finally we prove transplanting theorems à la Barth-Larsen for the Picard group of any isotropic grassmannian of lines and for the Neron-Severi group of some adjoint and coadjoint homogeneous spaces.     

Log-concavity of characteristic polynomials and the Bergman fan of matroids

In a recent paper, the first author proved the log-concavity of the coefficients of the characteristic polynomial of a matroid realizable over a field of characteristic 0, answering a long-standing conjecture of Read in graph theory. We extend the proof to all realizable matroids, making progress towards a more general conjecture of Rota?CHeron?CWelsh. Our proof follows from an identification of the coefficients of the reduced characteristic polynomial as answers to particular intersection problems on a toric variety. The log-concavity then follows from an inequality of Hodge type.     

Rational curves on K3 surfaces

We show that projective K3 surfaces with odd Picard rank contain infinitely many rational curves. Our proof extends the Bogomolov-Hassett-Tschinkel approach, i.e., uses moduli spaces of stable maps and reduction to positive characteristic.     

Differential equations with 2-term recursion

In this paper we consider linear differential equations with a recursion consisting of two terms. We consider these equations in positive characteristics and in characteristic zero. We will find a new proof for the Grothendieck conjecture for these equations. Supported by a grant from the DFG.     

Cayley’s and Holland’s Theorems for Idempotent Semirings and Their Applications to Residuated Lattices

We extend Cayley’s and Holland’s representation theorems to idempotent semirings and residuated lattices, and provide both functional and relational versions. Our analysis allows for extensions of the results to situations where conditions are imposed on the order relation of the representing structures. Moreover, we give a new proof of the finite embeddability property for the variety of integral residuated lattices and many of its subvarieties.     

Relevement Des Formes Modulaires De Picard

We prove a lifting theorem (from any characteristic p to characteristiczero) for holomorphic Picard modular forms of any weight andlevel. The proof makes use of the knowledge of the geometryof the toroidal compactification of the Picard modular surface.     

Automorphic equivalence in the varieties of representations of Lie algebras

Abstract

In this paper, we consider the wide class of subvarieties of the variety of all representation of Lie algebras over a field k of characteristic 0. We study the relation between the geometric equivalence and automorphic equivalence of the representations from these subvarieties.     

Irreducibility of the Igusa tower

We shall give a simple （basically） the Igusa tower over Shimura varieties of PEL purely in characteristic p proof of the irreducibility of type. Our result covers Shimura variety of type A and type C classical groups, in particular, the Siegel modular varieties, the Hilbert-Siegel modular varieties, Picard surfaces and Shimura varieties of inner forms of unitary and symplectic groups over totally real fields.     

Perverse coherent sheaves and the geometry of special pieces in the unipotent variety

Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U⊂X be an open set whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent intermediate extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent intermediate extension functor.Under suitable hypotheses, we introduce a construction (called “S₂-extension”) in terms of perverse coherent sheaves of algebras on X that takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical “S₂-ification” of appropriate X. The construction also has applications to the “Macaulayfication” problem, and it is particularly well-behaved when X is Gorenstein.Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown for the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S₂-extension to give a uniform construction of the desired variety.     

Which algebraic groups are Picard varieties?

We show that every connected commutative algebraic group over an algebraically closed field of characteristic 0 is the Picard variety of some projective variety having only finitely many non-normal points.In contrast,no Witt group of dimension at least 3 over a perfect field of prime characteristic is isogenous to a Picard variety obtained by this construction.     

Differential equations with 2-term recursion

In this paper we consider linear differential equations with a recursion consisting of two terms. We consider these equations in positive characteristics and in characteristic zero. We will find a new proof for the Grothendieck conjecture for these equations. Received: 20 June 1997/ Revised Version: 2 October 1997     

Décompte dans une conjecture de Lang

Faltings has proven the following conjecture of Lang: if A is an abelian variety over a number field and X any subvariety then all rational points of X lie on a finite number N of translates, contained in X, of abelian subvarieties of A. We provide an upper bound for N whose main feature is uniformity in X since it does not depend on the height of X. Moreover, the bound is completely explicit. Together with the result of a previous paper, the heart of the proof is a suitable generalization of Mumford’s theorem for curves.

---

Décompte dans une conjecture de Lang

---

Oblatum 20-III-2000 & 3-V-2000?Published online: 16 August 2000     

Periodic points, linearizing maps, and the dynamical Mordell-Lang problem

Under suitable hypotheses, we prove a dynamical version of the Mordell-Lang conjecture for subvarieties of quasiprojective varieties X, endowed with the action of a morphism . We also prove a version of the Mordell-Lang conjecture that holds for any endomorphism of a semiabelian variety. We use an analytic method based on the technique of Skolem, Mahler, and Lech, along with results of Herman and Yoccoz from nonarchimedean dynamics.     

Iterative processes related to Riordan arrays: The reciprocation and the inversion of power series

We point out how the Banach Fixed Point Theorem, together with the Picard successive approximation methods yielded by it, allows us to treat some mathematical methods in combinatorics. In particular we get, in this way, a proof of and an iterative algorithm for deriving the Lagrange Inversion Formula.