-dimensions of algebraic surfaces and numerically effective divisors
Let be a Fano manifold with different from the projective space such that any two surfaces in have proportional fundamental classes in . Let be a surjective holomorphic map from a projective variety . We show that all deformations of with and fixed, come from automorphisms of . The proof is obtained by studying the geometry of the integral varieties of the multi-valued foliation defined by the variety of minimal rational tangents of .
We use orbifold structures to deduce degeneracy statements for holomorphic maps into logarithmic surfaces. We improve former results in the smooth case and generalize them to singular pairs. In particular, we give applications on nodal surfaces and complements of singular plane curves.