Eisenman Intrinsic Measures and Algebraic Invariants

We generalize the Sakai theorem that says that every complex algebraic manifold of general type is measure hyperbolic. We introduce the notion of k-measure hyperbolicity for every Eisenman k-measure and, following Sakai, we consider an analogue k̄k of the Kodaira logarithmic dimension which construction uses logarithmic k-forms. We show that a complex algebraic manifold is k-measure hyperbolic if k̄k(X) = dimX.