Asymptotics of quantized barycenters of lattice polytopes with applications to algebraic geometry

TL;DR

This paper studies the asymptotic behavior of quantized barycenters of lattice polytopes, establishing their expansion, and applies these results to algebraic geometry, particularly in stability thresholds and invariants of toric varieties.

Contribution

It provides the first complete asymptotic expansion for quantized barycenters of lattice polytopes and links these to algebraic geometric invariants, advancing understanding of stability in toric varieties.

Findings

Established existence of asymptotic expansion and determined first two terms.

Derived all terms of the expansion for Delzant polytopes using mixed volumes and invariants.

Showed colinearity of barycenters for reflexive polytopes and polygons.

Abstract

This article addresses a combinatorial problem with applications to algebraic geometry. To a convex lattice polytope $P$ and each of its integer dilations $kP$ one may associate the barycenter of its lattice points. This sequence of $k$-quantized barycenters converge to the (classical) barycenter of the polytope considered as a convex body. A basic question arises: is there a complete asymptotic expansion for this sequence? If so, what are its terms? This article initiates the study of this question. First, we establish the existence of such an expansion as well as determine the first two terms. Second, for Delzant lattice polytopes we use toric algebra to determine all terms using mixed volumes of virtual rooftop polytopes, or alternatively in terms of higher Donaldson--Futaki invariants. Third, for reflexive polytopes we show the quantized barycenters are colinear to first order, and actually colinear in the case of polygons. The proofs use Ehrhart theory, convexity arguments, and toric algebra. As applications we derive the complete asymptotic expansion of the Fujita--Odaka stability thresholds $\delta_k$ on arbitrary polarizations on (possibly singular) toric varieties. In fact, we show they are rational functions of $k$ for sufficiently large $k$. This gives the first general result on Tian's stabilization problem for $\delta_k$-invariants for (possibly singular) toric Fanos: $\delta_k$ stabilize in $k$ if and only if they are all equal to $1$, and when smooth if and only if asymptotically Chow semistable. We also relate the asymptotic expansions to higher Donaldson--Futaki invariants of test configurations motivated by Ehrhart theory, and unify in passing previous results of Donaldson, Ono, Futaki, and Rubinstein--Tian--Zhang.