Complexifications of nonnegatively curved manifolds
Complexity of intersections of real quadrics and topology of symmetric determinantal varieties
Computing amoebas.
Construction d'hypersurfaces réelles
Cycles on algebraic models of smooth manifolds
Every compact smooth manifold is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of . We study modulo 2 homology classes represented by algebraic subsets of , as runs through the class of all algebraic models of . Our main result concerns the case where is a spin manifold.
Description of algebraically constructible functions.
Divisors on real curves.
Equivariant virtual Betti numbers
We define a generalised Euler characteristic for arc-symmetric sets endowed with a group action. It coincides with the Poincaré series in equivariant homology for compact nonsingular sets, but is different in general. We put emphasis on the particular case of , and give an application to the study of the singularities of Nash function germs via an analog of the motivic zeta function of Denef and Loeser.
Erratum to : Morphisms, line bundles and moduli spaces in real algebraic geometry
Every connected sum of lens spaces is a real component of a uniruled algebraic variety
We show that any finite connected sum of lens spaces is diffeomorphic to a real component of a uniruled projective variety, and prove a conjecture of János Kollár.
Finiteness theorems on Blow-Nash triviality for real algebraic singularities
Gale duality for complete intersections
We show that every complete intersection defined by Laurent polynomials in an algebraic torus is isomorphic to a complete intersection defined by master functions in the complement of a hyperplane arrangement, and vice versa. We call systems defining such isomorphic schemes Gale dual systems because the exponents of the monomials in the polynomials annihilate the weights of the master functions. We use Gale duality to give a Kouchnirenko theorem for the number of solutions to a system of master...
Géométrie algébrique réelle et composantes connexes
Géométrie réelle des dessins d’enfant
À tout dessin d’enfant est associé un revêtement ramifié de la droite projective complexe , non ramifié en dehors de 0, 1 et l’infini. Cet article a pour but de décrire la structure algébrique de l’image réciproque de la droite projective réelle par ce revêtement, en termes de la combinatoire du dessin d’enfant. Sont rappelées en annexe les propriétés de la restriction de Weil et des dessins d’enfants utilisées.
Géométrie réelle des dessins d’enfant : une étude des composantes irréductibles
Dans cet article nous nous intéressons aux propriétés des composantes irréductibles associées à la géométrie réelle d’un dessin d’enfant. Plus précisément, nous étudions les composantes irréductibles de la courbe dont l’ensemble des points réels est l’image réciproque de par une fonction de Belyi d’un dessin d’enfant.
Halves of a real Enriques surface.
Homology classes of real algebraic sets
There is a large research program focused on comparison between algebraic and topological categories, whose origins go back to 1952 and the celebrated work of J. Nash on real algebraic manifolds. The present paper is a contribution to this program. It investigates the homology and cohomology classes represented by real algebraic sets. In particular, such classes are studied on algebraic models of smooth manifolds.
Inflection points on real plane curves having many pseudo-lines.
Inner metric properties of 2-dimensional semi-algebraic sets.
We consider 2-dimensional semialgebraic topological manifolds from the differentialgeometric point of view. Curvatures at singularities are defined and a Gauss-Bonnet formula holds. Moreover, Aleksandrov's axioms for an intrinsic geometry of surfaces are fulfilled.
Lefschetz Fibrations and real Lefschetz fibrations
This note is based on the lectures that I have given during the winter school Winter Braids IV, School on algebraic and topological aspects of braid groups held in Dijon on 10 - 13 February 2014. The aim of series of three lectures was to give an overview of geometrical and topological properties of 4-dimensional Lefschetz fibrations. Meanwhile, I could briefly introduce real Lefschetz fibrations, fibrations which have certain symmetry, and could present some interesting features of them.This note...