A lower bound for polynomial volume growth of automorphisms of zero entropy · arXivDesk
2604.21398Apr 23, 202630 pages, 2 tables, title changed, we prove Conjecture 1.9 (Lower Bound) in the first version by introducing dynamical intersection polynomials whose total degree equals plov(f) - d; any comments are very welcome!
A lower bound for polynomial volume growth of automorphisms of zero entropy
Let X be a normal projective variety of dimension d, and let f be a zero-entropy automorphism of X. Denote by k the first-degree growth rate of f
Nearby in the stack
, so that
°1(fn)≍nk
. We prove the sharp lower bound for the polynomial volume growth
plov(f)
of
f
:
plov(f)≥d+4k(k+2),
equivalently giving a sharp lower bound on the Gelfand--Kirillov dimension of the associated twisted homogeneous coordinate ring. This improves previous lower bounds of Keeler and of Lin--Oguiso--Zhang. In the proof, we introduce the notion of dynamical intersection polynomials and give a new characterization of
plov(f)
in terms of non-vanishing of intersection numbers. We also establish a gap principle for polynomial volume growth: for every fixed dimension
d≥4
, either
plov(f)=d2
, or
plov(f)≤d(d−2)+2⌊d/4⌋
. This reveals a new rigidity phenomenon for zero-entropy automorphisms. As an application, in dimension
4
we determine all possible values of
plov
, thereby extending the results of Artin--Van den Bergh for surfaces and Lin--Oguiso--Zhang for threefolds.