Compositio Mathematica

Research Article

Polarized endomorphisms of uniruled varieties. With an appendix by Y. Fujimoto and N. Nakayama

De-Qi Zhanga1

a1 Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore (email:


We show that polarized endomorphisms of rationally connected threefolds with at worst terminal singularities are equivariantly built up from those on xs211A-Fano threefolds, Gorenstein log del Pezzo surfaces and xs21191. Similar results are obtained for polarized endomorphisms of uniruled threefolds and fourfolds. As a consequence, we show that every smooth Fano threefold with a polarized endomorphism of degree greater than one is rational.

(Received November 17 2008)

(Accepted March 04 2009)

(Online publication December 21 2009)

2000 Mathematics Subject Classification

  • 14E20;
  • 14J45;
  • 14E08;
  • 32H50


  • polarized endomorphism;
  • uniruled variety;
  • rationality of variety


This project is supported by an Academic Research Fund of NUS.

Dedicated to Professor M. Miyanishi on the occasion of his seventieth birthday