Exponential rarefaction of real curves with many components

Details Exponential rarefaction of real curves with many components Exponential rarefaction of real curves with many components

2010-05-18

arxiv.org/abs/1005.3228v1

Publ. Math. Inst. Hautes \'Etudes Sci. 113 (2011) 69-96

Mathematics

Algebraic Geometry

21 pages

Scientific paper

Given a positive real Hermitian holomorphic line bundle L over a smooth real projective manifold X, the space of real holomorphic sections of the bundle L^d inherits for every positive integer d a L^2 scalar product which induces a Gaussian measure. When X is a curve or a surface, we estimate the volume of the cone of real sections whose vanishing locus contains many real components. In particular, the volume of the cone of maximal real sections decreases exponentially as d grows to infinity.

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