Exponential rarefaction of real curves with many components

Damien Gayet (1) , Jean-Yves Welschinger (2)

1 ICJ - Institut Camille Jordan
2 AGL - Algèbre, géométrie, logique

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.

Mots clés: real maximal curves; large deviations; positive currents

Domaines: Géométrie algébrique [math.AG]

Damien Gayet, Jean-Yves Welschinger. Exponential rarefaction of real curves with many components. Publ. Math. Inst. Hautes Études Sci., 2011, 113, pp.69-96. ⟨10.1007/s10240-011-0033-3⟩. ⟨hal-00484611⟩