Exponential rarefaction of real curves with many components

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 inherits for every d ∈ N∗ a L 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. Mathematics subject classification 2010: 14P25, 32U40, 60F10 Introduction Let (X, cX) be a smooth real projective manifold of dimension n and (L, cL) π → (X, cX) be a real ample holomorphic line bundle. In particular, cX and cL are antiholomorphic involutions on X and L respectively, such that cX ◦ π = π ◦ cL. Let h be a real Hermitian metric on (L, cL) with positive curvature ω. It induces a Kähler structure on (X, cX). For every nonnegative integer d, this metric induces a Hermitian metric h on L and then a L-Hermitian product on the complex vector space H(X,L) of holomorphic sections of L. This product is defined by (σ, τ) ∈ H(X,L) × H(X,L) 7→ 〈σ, τ〉 = ∫ X h(σ, τ)dx ∈ C, where dx = ω/ ∫ X ω is the normalized volume induced by the Kähler form. Let RH(X,L) be the space of real sections {σ ∈ H(X,L) | cL ◦ σ = σ ◦ cX} and ∆k ⊂ H(X,L) (resp. R∆k ⊂ RH(X,L)) be the discriminant locus (resp. its real part), that is the set of sections which do not vanish transversally. For every σ ∈ H(X,L) \ {0}, denote by Cσ = σ−1(0) the vanishing locus of σ and when σ is real, by RCσ its real part. The divisor Cσ is smooth whenever σ ∈ RH(X,L)\R∆d. In this case, we denote by b0(σ) = b0(RCσ) the number of connected components of RCσ. 0.1 Real projective surfaces When X is two-dimensional, we know from Harnack-Klein inequality [9], [11] that b0(RCσ) ≤ g(Cσ) + 1, where equality holds for the so-called maximal curves. Here, the genus g(Cσ) of these smooth curves Cσ gets computed by the adjunction formula and equals g(Cσ) = 1 2 (dL−dc1(X).L+2), where c1(X) denotes the first Chern class