As an application of a recent characterization of complete flag manifolds as Fano manifolds having only P 1 -bundles as elementary contractions, we consider here the case of a Fano manifold X of Picard number one supporting an unsplit family of rational curves whose subfamilies parametrizing curves through a fixed point are rational homogeneous, and we prove that X is homogeneous. In order to do this, we first study minimal sections on flag bundles over the projective line, and discuss how Grothendieck's theorem on principal bundles allows us to describe a flag bundle upon some special sections.
Abstract. A Fano manifold X with nef tangent bundle is of Flag-Type if it has the same kind of elementary contractions as a complete flag manifold. In this paper we present a method to associate a Dynkin diagram D(X) with any such X, based on the numerical properties of its contractions. We then show that D(X) is the Dynkin diagram of a semisimple Lie group. As an application we prove that Campana-Peternell conjecture holds when X is a Flag-Type manifold whose Dynkin diagram is An, i.e. we show that X is the variety of complete flags of linear subspaces in P n .
In this work we deal with vector bundles of rank two on a Fano manifold X with second and fourth Betti numbers equal to one. We study the nef and pseudoeffective cones of the corresponding projectivizations and how these cones are related to the decomposability of the vector bundle. As consequences, we obtain the complete list of P 1 -bundles over X that have a second P 1 -bundle structure, classify all the uniform rank two vector bundles on this class of Fano manifolds and show the stability of indecomposable Fano bundles (with one exception on P 2 ).
The above theorem is proved in [26,5]; the two cases appearing in the theorem are called A 1 and A 2 , respectively, depending on the type of singularity the variety Y has along Z. The A 2 case is dealt with in codimension less than or equalSince is an isomorphism and T e and De are injection and projection, respectively, we see that rk U ¼ 2n À 2 and hence (ii) implies (iii). Ã Ã 1 T X 1 is a quotient of T X via the tangent map T 1 : T X ! Ã T X 1 . Moreover by [16] the divisor ÀK X 1 is 2 -ample. Therefore 2 : X 1 ! Y satises the conditions of Theorem 4.4 and ðX 1 =Y Þ < ðX=Y Þ, so the second part of the lemma follows by induction with respect to . Ã
We present here some conjectures on the diagonalizability of uniform principal bundles on rational homogeneous spaces, that are natural extensions of classical theorems on uniform vector bundles on the projective space, and study the validity of these conjectures in the case of classical groups.
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