Abstract
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 $$1\le k\le \text {dim}(X)-1$$ . We study the properties of $$X$$ . In particular 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, $$\text {dim}(X)\ge 3$$ 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.
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