Durfee's conjecture on the signature of smoothings of surface singularities

In 1978 Durfee conjectured various inequalities between the signature σ and the geometric genus pg of a normal surface singularity. Since then a few counter examples have been found and positive results established in some special cases. We prove a 'strong' Durfee-type inequality for any smoothing of a Gorenstein singularity, provided that the intersection form of the resolution is unimodular. We also prove the conjectured 'weak' in- equality for all hypersurface singularities and for sufficiently large multiplicity strict complete intersec- tions. The proofs establish general inequalities valid for any numerically Gorenstein normal surface singularity. © 2017 Société Mathématique de France. Tous droits réservés

On the cohomology of pseudoeffective line bundles

The goal of this survey is to present various results concerning the cohomology of pseudoeffective line bundles on compact K{\"a}hler manifolds, and related properties of their multiplier ideal sheaves. In case the curvature is strictly positive, the prototype is the well known Nadel vanishing theorem, which is itself a generalized analytic version of the fundamental Kawamata-Viehweg vanishing theorem of algebraic geometry. We are interested here in the case where the curvature is merely semipositive in the sense of currents, and the base manifold is not necessarily projective. In this situation, one can still obtain interesting information on cohomology, e.g. a Hard Lefschetz theorem with pseudoeffective coefficients, in the form of a surjectivity statement for the Lefschetz map. More recently, Junyan Cao, in his PhD thesis defended in Grenoble, obtained a general K{\"a}hler vanishing theorem that depends on the concept of numerical dimension of a given pseudoeffective line bundle. The proof of these results depends in a crucial way on a general approximation result for closed (1,1)-currents, based on the use of Bergman kernels, and the related intersection theory of currents. Another important ingredient is the recent proof by Guan and Zhou of the strong openness conjecture. As an application, we discuss a structure theorem for compact K{\"a}hler threefolds without nontrivial subvarieties, following a joint work with F.Campana and M.Verbitsky. We hope that these notes will serve as a useful guide to the more detailed and more technical papers in the literature; in some cases, we provide here substantially simplified proofs and unifying viewpoints.Comment: 39 pages. This survey is a written account of a lecture given at the Abel Symposium, Trondheim, July 201

Interacting Preformed Cooper Pairs in Resonant Fermi Gases

We consider the normal phase of a strongly interacting Fermi gas, which can have either an equal or an unequal number of atoms in its two accessible spin states. Due to the unitarity-limited attractive interaction between particles with different spin, noncondensed Cooper pairs are formed. The starting point in treating preformed pairs is the Nozi\`{e}res-Schmitt-Rink (NSR) theory, which approximates the pairs as being noninteracting. Here, we consider the effects of the interactions between the Cooper pairs in a Wilsonian renormalization-group scheme. Starting from the exact bosonic action for the pairs, we calculate the Cooper-pair self-energy by combining the NSR formalism with the Wilsonian approach. We compare our findings with the recent experiments by Harikoshi {\it et al.} [Science {\bf 327}, 442 (2010)] and Nascimb\`{e}ne {\it et al.} [Nature {\bf 463}, 1057 (2010)], and find very good agreement. We also make predictions for the population-imbalanced case, that can be tested in experiments.Comment: 10 pages, 6 figures, accepted version for PRA, discussion of the imbalanced Fermi gas added, new figure and references adde

Sous-groupes alg\'ebriques du groupe de Cremona

We give a complete classification of maximal algebraic subgroups of the Cremona group of the plane and provide algebraic varieties that parametrize the conjugacy classes. ----- Nous donnons une classification compl\`ete des sous-groupes alg\'ebriques maximaux du groupe de Cremona du plan et explicitons les vari\'et\'es qui param\`etrent les classes de conjugaison.Comment: Text in French, Translated introduction, 35 pages, 1 figure, to appear in Transform. Group

Classification of K3-surfaces with involution and maximal symplectic symmetry

K3-surfaces with antisymplectic involution and compatible symplectic actions of finite groups are considered. In this situation actions of large finite groups of symplectic transformations are shown to arise via double covers of Del Pezzo surfaces. A complete classification of K3-surfaces with maximal symplectic symmetry is obtained.Comment: 26 pages; final publication available at http://www.springerlink.co

The masterpieces of John Forbes Nash Jr.

In this set of notes I follow Nash’s four groundbreaking works on real algebraic manifolds, on isometric embeddings of Riemannian manifolds and on the continuity of solutions to parabolic equations. My aim has been to stay as close as possible to Nash’s original arguments, but at the same time present them with a more modern language and notation. Occasionally I have also provided detailed proofs of the points that Nash leaves to the reader

Bad loci of free linear systems

The bad locus of a free linear system L on a normal complex projective variety X is defined as the set B(L) \subset X of points that are not contained in any irreducible and reduced member of L. In this paper we provide a geometric description of such locus in terms of the morphism defined by L. In particular, assume that dim X = 2 and L is the complete linear system associated to an ample and spanned line bundle. It is known that in this case B(L) is empty unless X is a surface. Then we prove that, when the latter occurs, B(L) is not empty if and only if L defines a morphism onto a two dimensional cone, in which case B(L) is the inverse image of the vertex of the cone

Ample vector bundles and del Pezzo manifolds

Let E be an ample vector bundle of rank r on a smooth projective manifold X of dimension n \geq r+3. Pairs (X,E) as above are investigated under the assumption that E has a regular section vanishing anlong a Fano manifold Z of indez dim(Z)-1 and Picard number \rho(Z) \geq 2