The effect of analytic morphisms on plane curve singularities, via direct and inverse images, is studied in terms of certain objects attached to the morphism, its jacobian, trunk and tangent map among them.

We investigate the asymptotic behavior of curvatures of the Weil-Petersson metric in Teichmüller space. We use a pointwise curvature estimate to study directions, in the tangent space, of extremely negative curvature and directions of asymptotically zero curvatures.

We consider rationally connected complex projective manifolds $M$ and show that their loop spaces—infinite dimensional complex manifolds--have properties similar to those of $M$. Furthermore, we give a finite dimensional application concerning holomorphic vector bundles over rationally connected complex projective manifolds.