On invariance and Ricci-flatness of Hermitian metrics on open manifolds

We discuss a technique to construct Ricci-flat hermitian metrics on complements of (some) anticanonical divisors of almost homogeneous manifolds and discuss when this metric is complete and K"ahler. This construction has a strong interplay with invariance groups of the same dimension as the manifold acting with an open orbit. Lie groups of this type we call divisorial. As an application we can describe compact manifolds admitting a divisorially invariant K"ahler metric on an open subset. Finally, we see a connection between the reducibility of the anticanonical divisor and the non-triviality of the K"ahler cone on the complement.