Exponential rarefaction of maximal real algebraic hypersurfaces

Abstract

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 (2011) 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 $d'\<d$. Such a statement is inspired by the recent work of Diatta and Lerario (2022).