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Algebraic equivalence of real algebraic cycles

Miguel Abánades, Wojciech Kucharz (1999)

Annales de l'institut Fourier

Given a compact nonsingular real algebraic variety we study the algebraic cohomology classes given by algebraic cycles algebraically equivalent to zero.

Algebraically constructible chains

Hélène Pennaneac'h (2001)

Annales de l’institut Fourier

We construct for a real algebraic variety (or more generally for a scheme essentially of finite type over a field of characteristic 0 ) complexes of algebraically and k - 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.

Amibes de variétés algébriques et dénombrement de courbes

Ilia Itenberg (2002/2003)

Séminaire Bourbaki

Les amibesdes variétés algébriques dans ( * ) n sont les images de ces variétés par l’application des moments Log : ( * ) n n , Log : ( z 1 , ... , z n ) ( log | z 1 | , ... , log | z n | ) . 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...

Approximation by continuous rational maps into spheres

Wojciech Kucharz (2014)

Journal of the European Mathematical Society

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.

Alexander Degtyarev, Vlatcheslav Kharlamov (1997)

Revista Matemática de la Universidad Complutense de Madrid

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.

Betti numbers of random real hypersurfaces and determinants of random symmetric matrices

Damien Gayet, Jean-Yves Welschinger (2016)

Journal of the European Mathematical Society

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...

Clifford’s Theorem for real algebraic curves

Jean-Philippe Monnier (2010)

Annales de l’institut Fourier

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

Wojciech Kucharz, Santiago R. Simanca (2010)

Annales de l’institut Fourier

We provide a simple characterization of codimension two submanifolds of n ( ) that are of algebraic type, and use this criterion to provide examples of transcendental submanifolds when n 6 . If the codimension two submanifold is a nonsingular algebraic subset of n ( ) whose Zariski closure in n ( ) is a nonsingular complex algebraic set, then it must be an algebraic complete intersection in n ( ) .

Complex orientation formulas for M -curves of degree 4 d + 1 with 4 nests

S.Yu. Orevkov (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

On démontre la formule d’orientations complexes pour les M -courbes dans P 2 de degré 4 d + 1 ayant 4 nids. Cette formule généralise celle pour les M -courbes à nid profond. C’est un pas vers la classification des M -courbes de degré 9 .

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