Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1

Hwang, Jun-Muk

Annales de l'Institut Fourier, Tome 57 (2007) no. 3, pp. 815-823.

Soit $X$ une variété de Fano avec $b_2=1$ différente de l’espace projectif et telle que tout couple de surfaces dans $X$ ont des classes fondamentales dans $H_4(X,\mathbf{C})$ proportionnelles. Soit $f:Y\to X$ une application surjective d’une variété projective $Y$ dans $X$. Nous montrons que toute déformation de $f$ de $Y$ dans $X$ (fixés), provient d’automorphismes de $X$. La preuve est obtenue en étudiant la géométrie des variétés intégrales du feuilletage multi-valué défini par la variété des vecteurs tangents des courbes rationnelles minimales de $X$.

Let $X$ be a Fano manifold with $b_2=1$ different from the projective space such that any two surfaces in $X$ have proportional fundamental classes in $H_4(X,\mathbf{C})$. Let $f:Y\to X$ be a surjective holomorphic map from a projective variety $Y$. We show that all deformations of $f$ with $Y$ and $X$ fixed, come from automorphisms of $X$. The proof is obtained by studying the geometry of the integral varieties of the multi-valued foliation defined by the variety of minimal rational tangents of $X$.

Classification : 14J45, 32H02
Keywords: minimal rational curves, Fano manifold, deformation of holomorphic maps
Mot clés : courbes rationnelles minimales, variété de Fano, déformation des applications holomorphes

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