Abstract :

Let X be a projective Fano manifold of Picard number one. There is a folklore that any non constant endomorphism of X is an isomorphism. In the first half of this article, we will prove the folklore conjecture when the co-tangent bundle of X is algebraically completely integrable system and the tangent bundle of X is not nef. In the second half of the talk, we will give example of a collection of projective Fano manifolds of Picard rank one (different from the moduli space of vector bundles on algebraic curves) whose co-tangent bundle is algebraically completely integrable system. The collection includes the smooth intersection of two quadrics in $P^{2g+1}$. In particular, the conjecture holds true for smooth intersection of two quadrics in $P^{2g+1}$.