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arXiv · 1303.3035

Lower estimates for the expected Betti numbers of random real hypersurfaces

Damien Gayet·Jean-Yves Welschinger
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Abstract

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 K\"ahlerian 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}).

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Damien Gayet, Jean-Yves Welschinger. 2013-03-12. Lower estimates for the expected Betti numbers of random real hypersurfaces. https://doi.org/10.1112/jlms%2Fjdu018

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