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Abstract: In Euclidean spaces, convolution with a smoothing kernel gives regularization for subharmonic functions, plurisubharmonic functions, and more generally, functions whose complex Hessian is in a convex cone. We wish to explore how to generalize these results on K"ahler manifolds. Very few results are known beyond the plurisubharmonic case, and the usual integration approach runs into serious difficulties. Instead, we use a sup-convolution to obtain regularization results for a general convex cone on compact Kahler manifolds with nonnegative bisectional curvature. This is based on joint work with Yulun Xu.