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Article Dans Une Revue Journal of the Institute of Mathematics of Jussieu Année : 2014

Expected topology of random real algebraic submanifolds

Damien Gayet

Résumé

Let X be a smooth complex projective manifold of dimension n equipped with an ample line bundle L and a rank k holomorphic vector bundle E. We assume that 0< k <=n, that X, E and L are defined over the reals and denote by RX the real locus of X. Then, we estimate from above and below the expected Betti numbers of the vanishing loci in RX of holomorphic real sections of E tensored with L^d, where d is a large enough integer. Moreover, given any closed connected codimension k submanifold S of R^n with trivial normal bundle, we prove that a real section of E tensored with L^d has a positive probability, independent of d, to contain around the square root of d^n connected components diffeomorphic to S in its vanishing locus.
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Dates et versions

hal-00846761 , version 1 (19-07-2013)
hal-00846761 , version 2 (14-04-2014)

Identifiants

Citer

Damien Gayet, Jean-Yves Welschinger. Expected topology of random real algebraic submanifolds. Journal of the Institute of Mathematics of Jussieu, 2014, pp.10.1017/S1474748014000115. ⟨10.1017/S1474748014000115⟩. ⟨hal-00846761v2⟩
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