Exponential rarefaction of maximal real algebraic hypersurfaces

DOI10.4171/JEMS/1311arXiv2009.11951OpenAlexW3088910616MaRDI QIDQ6349827

Publication date: 24 September 2020

Abstract: Given an ample real Hermitian holomorphic line bundle

L

over a real algebraic variety

X

, the space of real holomorphic sections of

L^{o t i m e s d}

inherits a natural Gaussian probability measure. We prove that the probability that the zero locus of a real holomorphic section

s

of

L^{o t i m e s d}

defines a maximal hypersurface tends to

0

exponentially fast as

d

goes to infinity. This extends to any dimension a result of Gayet and Welschinger valid for maximal real algebraic curves inside a real algebraic surface. The starting point is a low degree approximation property which relates the topology of the real vanishing locus of a real holomorphic section of

L^{o t i m e s d}

with the topology of the real vanishing locus a real holomorphic section of

L^{o t i m e s d^{'}}

for a sufficiently smaller

d^{'} < d

. Such a statement is inspired by a recent work of Diatta and Lerario.

Full work available at URL: https://doi.org/10.4171/jems/1311

Mathematics Subject Classification ID

Topology of real algebraic varieties (14P25) Hypersurfaces and algebraic geometry (14J70) Integral representations; canonical kernels (Szeg?, Bergman, etc.) (32A25)