Research of Endomorphisms on projective algebraic varieties

本文研究了光滑投射簇X的结构，其中X上存在一个非同构的满射全纯映射，称之为非平凡满射自同态。椭圆曲线，阿贝尔变种，复曲面变种等都是这些变种的典型例子。它的结构应该非常简单。在复动力系统中，主要研究了复射影空间和K3曲面上的非平凡满射自同态。我们的观点是从代数簇的分类理论出发研究簇本身的结构，而不是研究自同态f。我们得到了具有非负科代拉维数且允许非平凡满自同态的光滑射影3-折叠的完全分类。2002年，作者对这类品种进行了分类，但有一个例外，在与中山升的合作中得到了解决。

The purpose of this research is to study the structure of smooth projective varieties X admitting a non-isomorphic surjective holomorphic map from X to itself which is called a nontrivial surjective endomorphism. Elliptic curves, abelian varieties, toric varieties etc. are typical examples of such varieties. And it is expected to have the very simple structure. In complex dynamical systems, a nontrivial surjective endomorphism on complex projective spaces and K3 surfaces has been mainly studied. Our viewpoint is to study the structure of the variety itself in view of the classification theory of algebraic varieties, not the endomorphism f. We have obtained a complete classification of smooth projective 3-folds with non-negative Kodaira dimension admitting a nontrivial surjective endomorphism. In 2002, the author has almost classified such varieties except one case and such exception has been solved in the joint work with Noboru Nakayama.