Pseudo-effective and nef cones on spherical varieties

We show that nef cycle classes on smooth complete spherical varieties are effective, and the products of nef cycle classes are also nef. Let X be a smooth projective spherical variety such that its effective cycle classes of codimension k are nef, where 1<= k <= dim(X)-1. We study the properties of X. And we show that if X is a toric variety, then X is isomorphic to the product of some projective spaces; if X is toroidal, then X is isomorphic to a rational homogeneous space; if X is horospherical, dim(X)>= 3 and k=2, then effective divisors on X are nef; if X is horospherical and effective divisors on X are nef, then there is a morphism from X to a rational homogeneous space such that each fiber is isomorphic to the product of some horospherical varieties of Picard number one.