Existence of sections of line bundles over a toroidal group and its applications (Q1340620)

We consider the existence problem of sections of line bundles over a toroidal group. We give a partial answer in the general case, from which we can obtain a Lefschetz type theorem. Let \(R(X,L)\) be the graded ring associated to a line bundle \(L\) over a toroidal group \(X\). We also prove that the transcendence degree of the field of the homogeneous fractions of degree zero of \(R(X,L)\) is infinite if \(L\) has a positive definite hermitian form. Furthermore, the equalities of the Kodaira dimension, the numerical rank and the rank are proved.