On the cone of curves of an abelian variety

L etX be a smooth projective variety over the complex numbers and let N1(X )b e the real vector space of 1-cycles on X modulo numerical equivalence. As usual denote by NE (X )t he cone of curves on X, i.e. the convex cone in N1(X) generated by the effective 1-cycles. One knows by the Cone Theorem (4) that the closed cone of curves NE(X) is rational polyhedral whenever c1(X) is ample. For varieties X such that c1(X) is not ample, however, it is in general difficult to determine the structure of NE(X). The purpose of this paper is to study the cone of curves of abelian varieties. Specifically, the abelian varieties X are determined such that the closed cone NE(X )i s rational polyhedral. The result can also be formulated in terms of the nef cone of X or in terms of the semi-group of effective classes in the Neron-Severi group of X.

设\(X\)是复数域上的一个光滑射影簇，且设\(N_1(X)\)是\(X\)上模数值等价的\(1\)-圈的实向量空间。像往常一样，用\(NE(X)\)表示\(X\)上的曲线锥，即\(N_1(X)\)中由有效\(1\)-圈生成的凸锥。由锥定理（4）可知，当\(c_1(X)\)是丰富的时，曲线的闭锥\(NE(X)\)是有理多面体的。然而，对于\(c_1(X)\)不丰富的簇\(X\)，一般来说确定\(NE(X)\)的结构是困难的。本文的目的是研究阿贝尔簇的曲线锥。具体地，确定使得闭锥\(NE(X)\)是有理多面体的阿贝尔簇\(X\)。这个结果也可以用\(X\)的数值有效锥或者用\(X\)的内龙 - 塞韦里群中的有效类半群来表述。