Soient k un corps de nombres, p un nombre premier, et L la p-tour de Hilbert de k; G = Gal(L/k). Dans ce papier, on se propose de donner un raffinement du critère de Golod-Safarevic appliqué à G, construit en filtrant le groupe de nombres Λ = {x ∈ k ×/(x) = }. Ce raffinement permettra alors de montrer qu’un corps quadratique réel dont le groupe des classes contient un sous-groupe de la forme (4, 4, 4, 4), possède une 2-tour de Hilbert infinie.

We use the method of Scholz and Reichardt and a transfer principle from finite fields to pseudo finite fields in order to prove the following result. THEOREM Let G be a group of order ln, where l is a prime number. Let K 0be either a finite field with |K 0| > l 4n+4or a pseudo finite field. Suppose that l ≠ char(K 0) and that K 0does not contain the root of unity ζl of order l. Let K = K 0(t), with t transcendental over K 0. Then K has a Galois extension L with the following properties: (a) (L/K) ≅ G; (b) L/K 0is a regular extension; (c) genus(L) < ; (d) K 0[t] has exactly n prime ideals which ramify in L; the degree of each of them is [K 0 : K 0]; (e) (t)∞totally decomposes in L; (f) L = K(x), withand deg(ai (t)) < deg(a 1(t)) for i = 1,…,ln .

The graphs that arise from the gradients of solutions u of the homogeneous complex Monge-Ampère equation are characterized in terms of the natural symplectic structure on the cotangent bundle. This characterization is invariant under symplectic biholomorphisms. Using the symplectic structures we construct symmetries (to be called Lempert transformations) for real valued functions u which are absolutely continuous on lines. We then use these symmetries to generate interesting solutions to the homogeneous complex Monge-Ampère equation and to transform the Poincaré-Lelong equation and the ∂-equation. An example of Lempert transform is given and the main theorem is applied to prove regularity results for exterior nonlinear Dirichlet problem for the homogeneous complex Monge-Ampère equation.

We show that the image of the moduli space of stable bundles on an Enriques surface by the pull back map is a Lagrangian subvariety in the moduli space of stable bundles, which is a symplectic variety, on the covering K3 surface. We also describe singularities and some other features of it.

Let f: M → M′ be a real analytic CR mapping between hyper-surfaces with f(p) = q, where p ∈ M and q ∈ M′. In this paper, the relation between the type at p and the one at q is considered. As a corollary of the type condition theorem (Theorem 1.1), a classification theorem, which states that under certain type condition, any real analytic CR mapping as above is constant, is proved.

Making use of 1-forms and geometric inequalities we prove the rigidity property of complete submanifolds Mn with parallel mean curvature normal in a complete and simply connected Riemannian (n+p) -manifold Nn+p with positive sectional curvature. For given integers n, p and for a nonnegative constant H we find a positive number T(n,p) ∈ (0,1) with the property that if the sectional curvature of N is pinched in [T(n,p), 1], and if the squared norm of the second fundamental form is in a certain interval, then Nn+p is isometric to the standard unit (n + p)-sphere. As a consequence, such an M is congruent to one of the five models as seen in our Main Theorem.

We classify smooth Fano 4-folds with second betti number b 2 = 2, possesing adjunction theoretic scroll structure. These manifolds occur to be projectivisations of coherent sheaves over Fano manifolds. The paper deals mainly with projectivisations of non-locally free rank 2 Fano sheaves over Fano 3-folds with b 2 = 1, using Mori theory. By the way, we classify nef and big rank 2 bundles over P2 with the first Chern class (3).

Consider a non-Gaussian SαS process X = {X(t); t ∈ T} which is expressed as a canonical representation , and is continuous in probability. If X is n-ple Markov, then X has determinism of dimension n + 1. That is, any SαS process having the same (n + l)-dimensional distributions with X is identical in law with X.

In this paper we prove Pesin’s entropy formula for general C 2 (or C 1+α) (non-invertible) endomorphisms of a compact manifold preserving a smooth measure.