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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2024_3656_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2024_3656_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation>-invariants for (possibly singular) toric Fanos: <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2024_3656_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> stabilize in k if and only if they are all equal to 1, and when smooth if and only if asymptotically Chow semistable (a condition stronger than existence of a Kähler–Einstein metric). 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, Futaki, Ono, Sano, and Rubinstein–Tian–Zhang on existence of canonical Kähler metrics, obstructions, and stability thresholds.

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

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发表日期：

2025年1月18日

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Chenzi Jin,
Yanir A. Rubinstein

摘要

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 $\delta _k$ $δ_{k}$ 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 $\delta _k$ $δ_{k}$ -invariants for (possibly singular) toric Fanos: $\delta _k$ $δ_{k}$ stabilize in k if and only if they are all equal to 1, and when smooth if and only if asymptotically Chow semistable (a condition stronger than existence of a Kähler–Einstein metric). 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, Futaki, Ono, Sano, and Rubinstein–Tian–Zhang on existence of canonical Kähler metrics, obstructions, and stability thresholds.