发表机构
Kspectra Research Inc.（Kspectra研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究正有限指数和对数满足广义爱因斯坦方程的条件，分类了二元洛朗多项式，证明了环面Kähler-Einstein流形的刚性，并刻画了有限指数族，解决了相关猜想。
AI 中文摘要
我们研究$\u211d^d$上正有限指数和的对数$\u03c8$何时满足$\u2207^2\u03c8=C\bumpe{\u27e8b,\u03b8\u27e9-\u03bb\u03c8}$。这是由指数映射诱导到射影空间的度量的Kähler-Einstein方程，对于自然指数族，它也是Jeffreys先验成为Diaconis--Ylvisaker共轭先验的条件。首先，我们分类了具有单模支撑且满足Di Scala和Sombra的广义爱因斯坦条件的二元洛朗多项式：在单位元和单项式坐标变换下，它们是仿射三项式的幂或两个独立二项式的幂的乘积。其次，我们证明在任意维数下，具有射影诱导的Kähler-Einstein度量的光滑紧致环面流形是带有匹配倍数的Fubini--Study度量的射影空间的乘积，由完全Veronese--Segre系统在自同构意义下浸入。这证明了此类度量的齐性猜想在紧致环面情形，以及Manno和Salis猜想的不动点芽形式和单叶形式。第三，在没有格点或理性假设的情况下，满足该方程的有限支撑指数族，在统计量的仿射变换下，恰好是具有相同类别与试验次数比的多项式族的乘积；这解决了Casalis问题在有限支撑情形。
英文摘要
We study when the logarithm $ψ$ of a positive finite exponential sum on $\mathbb{R}^d$ satisfies $\det\nabla^2ψ=C\exp(\langle b,θ\rangle-λψ)$. This is the Kähler--Einstein equation for metrics induced by exponential maps into projective space and, for natural exponential families, the condition that the Jeffreys prior be a Diaconis--Ylvisaker conjugate prior. First, we classify the bivariate Laurent polynomials with unimodular support satisfying the generalized Einstein condition of Di Scala and Sombra: up to units and monomial changes of coordinates, they are powers of an affine trinomial or products of powers of two independent binomials. Second, we show in every dimension that a smooth compact toric manifold with a projectively induced Kähler--Einstein metric is a product of projective spaces with matched multiples of the Fubini--Study metrics, immersed by a complete Veronese--Segre system up to automorphisms. This proves the compact toric case of the homogeneity conjecture for such metrics and the fixed-point germ and univalent forms of a conjecture of Manno and Salis. Third, without lattice or rationality assumptions, the finite-support exponential families satisfying the equation are, up to affine changes of statistic, exactly the products of multinomial families with a common ratio of categories to trials; this settles the finite-support case of a question of Casalis.
