Fei Hu, Nanjing University
"An upper bound for polynomial volume growth or Gelfand–Kirillov dimension of automorphisms of zero entropy"
Let
X
be
a
smooth
complex
projective
variety
of
dimension
d
and
f
an
automorphism
of
X.
Suppose
that
the
pullback
f^*
of
f
on
the
real
Néron–Severi
space
N^1(X)_R
is
unipotent
and
denote
the
index
of
the
eigenvalue
1
by
k+1.
We
prove
an
upper
bound
for
the
polynomial
volume
growth
plov(f)
of
f,
or
equivalently,
for
the
Gelfand–Kirillov
dimension
of
the
twisted
homogeneous
coordinate
ring
associated
with
(X,
f),
as
follows:
plov(f)
\leq
(k/2
+
1)d.
Combining
with
the
inequality
k
\leq
2(d-1)
due
to
Dinh–Lin–Oguiso–Zhang,
we
obtain
an
optimal
inequality
that
plov(f)
\leq
d^2,
which
affirmatively
answers
questions
of
Cantat–Paris-Romaskevich
and
Lin–Oguiso–Zhang.
This
is
joint
work
with
Chen
Jiang.
MC 5417