A conic and an -quintic with a point at infinity.
A Formula for the Euler characteristic of a real algebraic manifold.
About the Euler-Poincaré characteristic of semi-algebraic sets defined with two inequalities.
We express the Euler-Poincaré characteristic of a semi-algebraic set, which is the intersection of a non-singular complete intersection with two polynomial inequalities, in terms of the signatures of appropriate bilinear symmetric forms.
Algebraic cycles and the classical groups II: quaternionic cycles.
Algebraic equivalence of real algebraic cycles
Given a compact nonsingular real algebraic variety we study the algebraic cohomology classes given by algebraic cycles algebraically equivalent to zero.
Algebraically constructible chains
We construct for a real algebraic variety (or more generally for a scheme essentially of finite type over a field of characteristic ) complexes of algebraically and - algebraically constructible chains. We study their functoriality and compute their homologies for affine and projective spaces. Then we show that the lagrangian algebraically constructible cycles of the cotangent bundle are exactly the characteristic cycles of the algebraically constructible functions.
Algebraically constructible functions
Algebraically constructible functions and signs of polynomials.
Amibes de variétés algébriques et dénombrement de courbes
Les amibesdes variétés algébriques dans sont les images de ces variétés par l’application des moments , . Des résultats obtenus par G. Mikhalkin montrent l’utilité des amibes pour l’étude des variétés algébriques réelles et complexes. Les amibes peuvent être déformées en des complexes polyédraux appelésvariétés algébriques tropicales. Cette déformation permet, en particulier, de calculer les invariants de Gromov-Witten du plan projectif et d’autres surfaces toriques en dénombrant des courbes...
An extremal property of Rokhlin' s inequality for real algebraic curves.
Approximation by continuous rational maps into spheres
Investigated are continuous rational maps of nonsingular real algebraic varieties into spheres. In some cases, necessary and sufficient conditions are given for a continuous map to be approximable by continuous rational maps. In particular, each continuous map between unit spheres can be approximated by continuous rational maps.
Around real Enriques surfaces.
We present a brief overview of the classification of real Enriques surfaces completed recently and make an attempt to systemize the known classification results for other special types of surfaces. Emphasis is also given to the particular tools used and to the general phenomena discovered; in particular, we prove two new congruence type prohibitions on the Euler characteristic of the real part of a real algebraic surface.
Behavior of Welschinger invariants under Morse simplifications
Betti numbers of random real hypersurfaces and determinants of random symmetric matrices
We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the Kählerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from...
Betti numbers of real elliptic surfaces. (Nombres de Betti des surfaces elliptiques réelles.)
Clifford’s Theorem for real algebraic curves
We establish, for smooth projective real curves, an analogue of the classical Clifford inequality known for complex curves. We also study the cases when equality holds.
Codimension two transcendental submanifolds of projective space
We provide a simple characterization of codimension two submanifolds of that are of algebraic type, and use this criterion to provide examples of transcendental submanifolds when . If the codimension two submanifold is a nonsingular algebraic subset of whose Zariski closure in is a nonsingular complex algebraic set, then it must be an algebraic complete intersection in .
Complete decomposability of quotients by complex conjugation for real complete intersection surfaces.
Complex arrangements and derivations.
Complex orientation formulas for -curves of degree with 4 nests
On démontre la formule d’orientations complexes pour les -courbes dans de degré ayant nids. Cette formule généralise celle pour les -courbes à nid profond. C’est un pas vers la classification des -courbes de degré .