Some results on compact Kähler surfaces with non-positive bisectional curvature

In 1961 L. Frankel conjectured that any compact Kähler manifold of positive bisectional curvature is biholomorphic to a projective space. This was solved by S. Mori in 1979 and Siu and Yau in 1980. N. Mok in 1988 gave a full solution to the generalized Frankel's conjecture and proposed a dual problem. The results in this paper are attempts to give modified solutions to this problem in complex dimension two.