Research of Endomorphisms on projective algebraic varieties

本文研究光滑射影变体X的结构，其中X到X本身有一个非同构满射全纯映射，称为非平凡满射自同态。椭圆曲线型、阿贝尔型、环面型等都是这类品种的典型代表。它的结构很简单。在复杂动力系统中，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变量的分类理论出发，而不是从自同态f的角度来研究变量本身的结构。我们得到了具有非平凡满射自同态的非负Kodaira维光滑投影3-fold的完全分类。在2002年，除了一个例外，作者几乎对这些品种进行了分类，并在与中山信郎的合作中解决了这一例外。

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.