In this paper we prove classification results for gradient shrinking Ricci solitons under two invariant conditions, namely nonnegative orthogonal bisectional curvature and weakly PIC1, without any curvature upper bound. New results on ancient solutions for the Ricci and Kähler-Ricci flow are obtained. Applications to Kähler manifolds with almost nonnegative orthogonal bisectional curvature are derived as consequences. The main new feature is that no curvature upper bound is assumed.