Polynomial log-volume growth in slow dynamics and the GK-dimensions of twisted homogeneous coordinate rings

Abstract

Twisted homogeneous coordinate rings are natural invariants associated to a projective variety X with an automorphism f . We study the Gelfand-Kirillov dimensions of these noncommutative algebras from the perspective of complex dynamics, by noticing that when X is a smooth complex projective variety, they essentially coincide with the polynomial logarithmic volume growth Plov(f ) of (X, f ). We formulate some basic dynamical properties about these invariants and study explicit examples. Our main results are new upper bounds and lower bounds of these invariants, in terms of the dimension of X (as well as other refinements). As an application, we completely determine Plov(f ) when dim X = 3.

Polynomial log-volume growth of quasi-unipotent automorphisms of abelian varieties (with an appendix in collaboration with Chen Jiang)

“…The most fascinating point of this dynamical invariant plov(f ) is its connection with the so-called Gelfand-Kirillov dimension GKdim B of the twisted homogeneous coordinate ring B associated with (X, f, O X (H X )), which will be discussed in the next section 3. This surprising connection was first noticed in [LOZ21].…”

“…Remark 1.2. As mentioned earlier, Theorem 1.1 has been proved by Lin, Oguiso, and Zhang for complex tori (see [LOZ21,Theorem 7.1]). Their proof is analytic and relies on calculations of differential forms; in particular, certain positivity of (1, 1)-forms is involved.…”

“…We interpret the top self-intersection number of ∆ n as the degree of the endomorphism φ O X (∆n) induced by the line bundle O X (∆ n ) on X using the Riemann-Roch theorem for abelian varieties. Then based on previous works of Keeler [Kee00] and Lin-Oguiso-Zhang [LOZ21], it suffices to evaluate the degree of the degree function deg(φ O X (∆n) ) as a polynomial in n, which can be done by an explicit calculation with the aid of Theorems 1.5 and 1.8 and Proposition A.2.…”

“…We now study algebraic dynamics of an automorphism f of zero entropy of a smooth projective variety X of dimension d over k. We give the definition of the polynomial log-volume growth plov(f ) of f and investigate several basic properties. We refer to [CPR21, DLOZ22,LOZ21] for recent advance on this subject (especially, when k = C).…”

“…Using this decomposition, the authors of [LOZ21] have obtained the following finiteness result on plov. See also [Kee00, Proof of Lemma 6.13].…”