2013/03/12 by Damien Gayet, Jean-Yves Welschinger
Mathematics · #Affine transformation #Algebraic Geometry and Number Theory #Ball (mathematics) #Betti number #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Hermitian matrix #Hypersurface #Manifold (fluid mechanics) #Real line #Real number #Upper and lower bounds #math.SG
paper · pdf · doi:10.1112/jlms/jdu018
19 pages
arxiv created 2013/03/12 · openalex publication_date 2014/05/20 · openalex created_date 2016/06/24 · arxiv updated 2017/05/17 · openalex updated_date 2026/08/05
We estimate from below the expected Betti numbers of real hypersurfaces taken at random in a smooth real projective n-dimensional manifold. These random hypersurfaces are chosen in the linear system of a large dth power of a real ample line bundle equipped with a Hermitian metric of positive curvature. As for the upper bounds that we recently established, these lower bounds read as a product of a factor which only depends on the dimension n of the manifold with the Kähler volume of its real locus R X and d n . Actually, any closed affine real algebraic hypersurface appears with positive probability as part of such random real hypersurfaces in any ball of R X of radius O ( 1 / d ) .