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).