Research of Endomorphisms on projective algebraic varieties

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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.