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A study of compact complex manifolds admitting non-isomorphic surjective endomorphisms

承认非同构满射自同态的紧复流形的研究

基本信息

批准号：
23540055
负责人：
FUJIMOTO Yoshio
金额：
$ 1.91万
依托单位：
Nara Medical University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2011
资助国家：
日本
起止时间：
2011 至 2013
项目状态：
已结题

项目摘要

We have studied structures of smooth projective 3-folds X with negative Kodaira dimension admitting non-isomorphic \'{e}tale endomorphisms. We can show that a suitable finite \'{e}tale covering X' of X is classified into 6 types. There exists some X where X' cannot be decomposed as a product of a uniruled surface and an elliptic curve. The contraction morphism associated to R is the inverse of a blow-up of another smooth 3-fold along an elliptic curve. The trouble is that R cannot necessarily be stabilized by a suitable power of f. Hence the minimal model program(MMP) cannot work so as to be compatible with endomorphisms. As a substitute, we consider the tower of non-isomorphic finite \'{e}tale covering of 3-folds (called ESP), on which MMP works well. Such a 3-fold X can be obtained as a succession of blow-ups along an elliptic curve from a characteristic ESP (called DESP). We can also give a simple criterion for certan conic bundles to have non-isomorphic \'{e}tale endomorphisms.

我们研究了具有负 Kodaira 维数的平滑射影 3 重 X 的结构，允许非同构的“故事自同态”。我们可以证明覆盖 X 的 X 的合适的有限“{e}故事”可分为 6 种类型。存在一些X，其中X'不能分解为无规曲面和椭圆曲线的乘积。与 R 相关的收缩态射是沿着椭圆曲线的另一个平滑 3 倍的放大的逆。问题是 R 不一定能通过合适的 f 次幂来稳定。因此最小模型程序（MMP）无法与自同态兼容。作为替代，我们考虑 3 重非同构有限 \'{e}tale 覆盖塔（称为 ESP），MMP 在该塔上运行良好。这样的 3 倍 X 可以通过特征 ESP（称为 DESP）沿椭圆曲线的一系列放大来获得。我们还可以给出一个简单的标准来判断某些圆锥丛是否具有非同构的自同态。