Inverse of abelian integrals and ramified Riemann domains (Q514347)

The Levi problem for ramified Riemann domains is considered: is any locally Stein domain over \(\mathbb C ^n\) Stein? It is known by Fornaess that it is not always the case. The author gives some sufficient conditions which ensure that the answer to the above question is ``yes''. He considers complex manifolds with global frames of the holomorphic cotangent bundles by closed abelian differentials and defines a scalar function which corresponds to the Hartogs radius in the case of domains in \(\mathbb C ^n\). A condition phrased in terms of this function leads to the positive answer to the Levi problem. A new proof of Behnke-Stein theorem is also given.