Bigness of tangent bundles and dynamical rigidity of Fano manifolds of Picard number 1 (with an appendix by Jie Liu)

Abstract

Let $f:X\to Y$ be a surjective morphism of Fano manifolds of Picard number 1 whose VMRTs at a general point are not dual defective. Suppose that the tangent bundle $T_X$ is big. We show that $f$ is an isomorphism unless $Y$ is a projective space. As applications, we study the bigness of the tangent bundles of complete intersections, del Pezzo manifolds, and Mukai manifolds, as well as their dynamical rigidity.

Mathematische Annalen (to appear). There is an important revision in the statement of Theorem 1.4. 24 pages with an appendix by Jie Liu