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

TL;DR

This work analyzes the asymptotics of quantized barycenters $Bc_k(P)$ of lattice polytopes, establishing a complete expansion framework that links discrete barycenters to the classical barycenter $Bc(P)$ and to stability invariants in toric geometry. It provides a general first-order expansion for arbitrary lattice polytopes and a full mixed-volume-based expansion in the Delzant (smooth) case, with refined first-order colinearity for reflexive polytopes and explicit formulas in two dimensions. These expansions yield complete asymptotics for the Fujita–Odaka $\delta_k$-invariants on toric (potentially singular) varieties and connect the coefficients to higher Donaldson–Futaki invariants via rooftop test configurations, unifying and extending prior results of Donaldson, Ono, Futaki, and Rubinstein–Tian–Zhang. The paper supplies a robust toolkit for understanding stability thresholds through discrete to continuous transitions, with practical computations validated through explicit polygon and higher-dimensional examples. The results illuminate stabilization phenomena, reveal rational dependence on the expansion parameter $k$ in many toric settings, and advance the program of complete asymptotics in algebro-geometric invariants.

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 $δ_k$ 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 $δ_k$-invariants for (possibly singular) toric Fanos: $δ_k$ 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.

Figures (9)

• Figure 1: The fan of the new toric manifold $\overline{{\mathcal{X}}}$. Note that $X$ corresponds to the subfan consisting of the cones in $M_{\mathbb R}\subseteq M_{\mathbb R}\times{\mathbb R}$. The cones above $M_{\mathbb R}$ are spanned by the cones in $M_{\mathbb R}$ and the ray generated by $(0,1)$, while the cones below $M_{\mathbb R}$ are spanned by the cones in $M_{\mathbb R}$ and the ray generated by $(v,-1)$.
• Figure 2: The projection $\pi:\overline{{\mathcal{X}}}\to{\mathbb P}^1$.
• Figure 3: The lattice polytope $P_i$: $u_i+C_i$ lattice points above each lattice point in $P$ with $i$-th coordinate equal to $u_i$. The dotted lines are fibers of $\pi:M\times{\mathbb R}\ni (u,h)\mapsto u\in M$.
• Figure 4: The polytope $P_1$ corresponding to ${\mathbb P}^2$ with $v_1=(1,0)$, $v_2=(0,1)$, $v_3=(-1,-1)$.
• Figure 5: The indexing of the edges and the lattice points of the polygon corresponding to the blow-up of ${\mathbb P}^1\times{\mathbb P}^1$.

Theorems & Definitions (77)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Definition 2.5
• Theorem 2.6
• Definition 2.7
• Theorem 2.8
• Definition 2.9
• Remark 2.10
• Theorem 2.11