Luis E. Solá Conde

Rationally connected foliations after Bogomolov and McQuillan

This paper is concerned with a sufficient criterion to guarantee that a given foliation on a normal variety has algebraic and rationally connected leaves. Following ideas from a preprint of Bogomolov-McQuillan and using recent works of Langer and Graber-Harris-Starr, we give a clean, short and simple proof of previous results. Apart from a new vanishing theorem for vector bundles in positive characteristic, our proof employs only standard techniques of Mori theory and does not make any reference to the more involved properties of foliations in characteristic p. We apply the result to show that Q-Fano varieties with unstable tangent bundles always admit a sequence of partial rational quotients naturally associated to the Harder-Narasimhan filtration.

On Manifolds Whose Tangent Bundle is Big and 1-Ample

A line bundle over a complex projective variety is called big and 1-ample if a large multiple of it is generated by global sections and a morphism induced by the evaluation of the spanning sections is generically finite and has at most 1-dimensional fibers. A vector bundle is called big and 1-ample if the relative hyperplane line bundle over its projectivisation is big and 1-ample. The main theorem of the present paper asserts that any complex projective manifold of dimension 4 or more, whose tangent bundle is big and 1-ample, is equal either to a projective space or to a smooth quadric. Since big and 1-ample bundles are ?almost? ample, the present result is yet another extension of the celebrated Mori paper ?Projective manifolds with ample tangent bundles? (Ann. of Math. 110 (1979) 593?606). The proof of the theorem applies results about contractions of complex symplectic manifolds and of manifolds whose tangent bundles are numerically effective. In the appendix we re-prove these results.

Fano manifolds whose elementary contractions are smooth P^1-fibrations: a geometric characterization of flag varieties

The present paper provides a geometric characterization of complete flag varieties for semisimple algebraic groups. Namely, if

Rational curves, Dynkin diagrams and Fano manifolds with nef tangent bundle

X

On rank

is a Fano manifold whose all elementary contractions are

2

\mathbb P^1

vector bundles on Fano manifolds

-fibrations then

A survey on the Campana-Peternell Conjecture

X

Existence of a virtual cathode close to a strongly electron emissive wall in low density plasmas

is isomorphic to the complete flag manifold

Non-equilibrium thermionic electron emission for metals at high temperatures

G/B