A classification theorem on Fano bundles

Muñoz, Roberto; Solá Conde, Luis E.; Occhetta, Gianluca

Annales de l'Institut Fourier, Tome 64 (2014) no. 1, pp. 341-373.

Dans cet article, on classifie les fibrés de Fano de rang deux $ℰ$ sur les variétés de Fano satisfaisant $H^2(X,\mathbb{Z})\cong H^4(X,\mathbb{Z})\cong \mathbb{Z}$. La classification est obtenue par le calcul des cônes nef et pseudoeffectif de la projectivation $\mathbb{P}\left(ℰ\right)$, ce qui nous permet d’obtenir des invariants cohomologiques de $X$ et $ℰ$. Comme un sous-produit, nous discutons des fibrés associés à Fano congruences de droites, montrant que leurs variétés de tangentes rationnelles minimales peuvent avoir plusieurs composants linéaires.

In this paper we classify rank two Fano bundles $ℰ$ on Fano manifolds satisfying $H^2(X,\mathbb{Z})\cong H^4(X,\mathbb{Z})\cong \mathbb{Z}$. The classification is obtained via the computation of the nef and pseudoeffective cones of the projectivization $\mathbb{P}\left(ℰ\right)$, that allows us to obtain the cohomological invariants of $X$ and $ℰ$. As a by-product we discuss Fano bundles associated to congruences of lines, showing that their varieties of minimal rational tangents may have several linear components.

Classification : 14M15, 14E30, 14J45
Keywords: vector bundles, Fano manifolds
Mot clés : fibrés vectorielles, variétés de Fano

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