Pré-Publication, Document De Travail Année : 2020

Exponential rarefaction of maximal real algebraic hypersurfaces

Résumé

Given an ample real Hermitian holomorphic line bundle $L$ over a real algebraic variety $X$, the space of real holomorphic sections of $L^{\otimes d}$ inherits a natural Gaussian probability measure. We prove that the probability that the zero locus of a real holomorphic section $s$ of $L^{\otimes d}$ 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 $L^{\otimes d}$ with the topology of the real vanishing locus a real holomorphic section of $L^{\otimes d'}$ for a sufficiently smaller $d'

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hal-02948676 , version 1 (24-09-2020)

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  • HAL Id : hal-02948676 , version 1

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Michele Ancona. Exponential rarefaction of maximal real algebraic hypersurfaces. 2020. ⟨hal-02948676⟩
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