Lefschetz type theorem

Summary Let X=Cⁿ / be a toroidal group of rank =n+m. If X is compact, then it is a complex torus. In the compact case, we have the theorem of Lefschetz which asserts that if L is a positive line bundle over a complex torus X, then gives an embedding of X for any integer l3. This theorem is generalized to noncompact toroidal groups in this paper. In fact, we prove the following: (I) In the case of rank =n+1, H⁰(X, (Ll)) gives an embedding of a toroidal group X for l3, if L is positive. (II) In the case of rank =n+m, 2m4 provided that a conjecture about the existence of nonzero section holds.