Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees
Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees
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DOI：
10.1353/ajm.1997.0033
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发表时间：
1996-10
影响因子：
1.7
通讯作者：
Y. Siu;Sai-Kee Yeung
中科院分区：
文献类型：
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作者：
Y. Siu;Sai-Kee Yeung
我们证明，对于从仿射复线到交换簇 A 的全纯映射 f 以及 A 中的充足除数 D，缺陷消失。该证明利用 f 图像的 k -jet 空间的 Zariski 闭包的平移不变性和黎曼·罗赫定理构造了一个非同零亚纯 k -jet 微分，其极除数由等于 pD 的除数主导，并且沿着 D 的 k -jet 空间消失到 q 阶，其中 p / q 小于规定的小正数。然后使用涉及除数为 D 的 theta 函数和对数导数引理的估计。我们还证明了一个点施瓦茨引理，它通过从仿射复线到紧复流形的全纯映射给出了回拉的消失，全纯射流微分在充足除数上消失。这个点施瓦茨引理是对格林和格里菲斯在对阿贝尔簇整个曲线上的布洛赫定理的另一种处理中所勾画的证明的一个陈述的轻微修改。还给出了点施瓦茨引理的对数极点情况。我们构造双曲超曲面的例子，其次数仅为其维数平方的 16 倍。
We prove that the defect vanishes for a holomorphic map f from the affine complex line to an abelian variety A and for an ample divisor D in A . The proof uses the translational invariance of the Zariski closure of the k -jet space of the image of f and the theorem of Riemann Roch to construct a nonidentically zero meromorphic k -jet differential whose pole divisor is dominated by a divisor equivalent to pD and which vanishes along the k -jet space of D to order q with p / q smaller than a prescribed small positive number. Then estimates involving the theta function with divisor D and the logarithmic derivative lemma are used. We also prove a pointwise Schwarz lemma which gives the vanishing of the pullback, by a holomorphic map from the affine complex line to a compact complex manifold, of a holomorphic jet differential vanishing on an ample divisor. This pointwise Schwarz lemma is a slight modification of a statement whose proof Green and Griffiths sketched in their alternative treatment of Bloch's theorem on entire curves in abelian varieties. The log-pole case of the pointwise Schwarz lemma is also given. We construct examples of hyperbolic hypersurface whose degree is only 16 times the square of its dimension.