On the topology of random real complete intersections

DOI10.1007/S12220-022-01092-XarXiv2109.13538MaRDI QIDQ6378762

Publication date: 28 September 2021

Abstract: Given a real projective variety

X

and

m

ample line bundles

L_{1}, d o t s L_{m}

on

X

also defined over

m a t h b b R

, we study the topology of the real locus of the complete intersections defined by global sections of

L_{1}^{o t i m e s d} o p l u s c d o t s o p l u s L_{m}^{o t i m e s d}

. We prove that the Gaussian measure of the space of sections defining real complete intersections with high total Betti number (for example, maximal complete intersections) is exponentially small, as

d

grows to infinity. This is deduced by proving that, with very high probability, the real locus of a complete intersection defined by a section of

L_{1}^{o t i m e s d} o p l u s d o t s o p l u s L_{m}^{o t i m e s d}

is isotopic to the real locus of a complete intersection of smaller degree.

Mathematics Subject Classification ID

Geometric probability and stochastic geometry (60D05) Kähler manifolds (32Q15) Topology of real algebraic varieties (14P25) Integral representations; canonical kernels (Szeg?, Bergman, etc.) (32A25)

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