Generalized Einstein Laurent polynomials, Toric Kähler-Einstein Rigidity, and Finite Exponential Families

Abstract

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.