Seminar

I will discuss (mostly) finite morphisms of projective manifolds, in particular deformation theory and positivity of the associated vector bundle and apply that in geometric situations. In particular the deformations of a surjective morphism f from a normal projective variety to a non-uniruled projective manifold is non-obstructed and all components of the Hom scheme are abelian varieties. More precisely, all small deformations of f come from automorphisms of some etale cover over
which f factors. An important role in the proof plays the rank (d-1) bundle E assoicated to a d-sheeted cover and its positivity properties. We show that E is ample if f is Galois, does not factor through anÊetale map and if the irreducible components of the ramification divisor are ample.