Lower estimates for the expected Betti numbers of random real hypersurfaces

Damien Gayet (1) , Jean-Yves Welschinger (1)

1 AGL - Algèbre, géométrie, logique

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 d-th 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 constant which only depends on the dimension n of the manifold with the Kählerian volume of its real locus RX and d^{n/2}. Actually, any closed affine real algebraic hypersurface appears with positive probability as part of such random real hypersurfaces in any ball of RX of radius O(d^{-1/2}).

Mots clés: random polynomial; ample line bundle; Real projective manifold; Betti numbers

Domaines: Géométrie symplectique [math.SG]; Géométrie algébrique [math.AG]

Damien Gayet, Jean-Yves Welschinger. Lower estimates for the expected Betti numbers of random real hypersurfaces. Journal of the London Mathematical Society, 2014, (2), 90 (no. 1), pp.105-120. ⟨10.1112/jlms/jdu018⟩. ⟨hal-00799627⟩