Volume 276, 2001
Number of automorphisms of
principally polarized abelian varieties
Explicit estimates on the size of the automorphism group of prin-
cipally polarized abelian varieties are given adapting methods of Catanese-
Schneider and Szabo for log varieties of general type.
There has been a considerable amount of work devoted to giving estimates
on the size of the automorphism groups of varieties of general type. The recent
articles [Catanese-Schneider95, Szab696, Xiao94/95] and the references contained
there provide ample material on this subject.
The purpose of this note is to point out that the methods of the above authors
and their predecessors can be used to obtain estimates in more general situations,
namely for log varieties of log general type. Note that among many others princi-
pally polarized abelian varieties belong to this class.
The first section contains the necessary modifications of the known arguments
to the log case. The main result is (1.3.1). The general results are applied for
principally polarized abelian varieties in the second section. The estimates are
clearly far from being sharp, but the idea of viewing a principally polarized abelian
log variety of general type may be worth mentioning.
AcKNOWLEDGEMENT. Most of the ideas behind the results in this note come
from other people. My modest claim of novelty is simply the realization that these
ideas apply under broader circumstences than previously used. In particular this
article benefited greatly from the papers of Catanese and Schneider, Szabo, and in
an indirect way from that of Huckleberry and Sauer. The results included in the
first section are due to these authors as much as or perhaps more than to me.
This paper would have never been written without Emma Previato's support.
I'd like to thank her for her interest and encouragement.
Thanks are also due to the referee whose comments improved the presentation.
DEFINITIONS AND NOTATION. Every object is defined over an algebraically
closed field of characteristic p
Mathematics Subject Classification.
14K99, 14J50, 14J29.
Supported in part by NSF Grant DMS-9818357.
2001 American l\Iatlwmatical Socit'ty