Problems in the Geometry of the Siegel-Jacobi Space

Yang, Jae-Hyun

doi:10.48550/arxiv.1910.06249

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The Moduli Of Abelian Varieties1

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2021

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“…From this, we can see that the Ricci curvature is non-positive and the metrics have constant scalar curvature. Using Yang's notation, this answers Question 3 of [Yan19] for the spaces 26 H 1,n , ds 2 1,n,A,B . It is also possible to compute the orthogonal anti-bisectional curvature of these spaces.…”

Section: The Moduli Of Abelian Varietiesmentioning

confidence: 82%

A Hall of Statistical Mirrors

Khan¹,

Zhang²

2021

Preprint

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The primary focus of information geometry are statistical manifolds, which are parametrized families of probability measures, induced with the Fisher-Rao metric and a pair of torsion-free conjugate connections. In recent work [ZK20], the authors considered parametrized probability distributions as partially-flat statistical manifolds admitting torsion and showed that there is a complex to symplectic duality on the tangent bundles of such manifolds, based on the dualistic geometry of the underlying manifold.In this paper, we explore this correspondence further in the context of Hessian manifolds, in which case the conjugate connections are both curvature-and torsion-free, and the associated dual pair of spaces are Kähler manifolds. We focus on several key examples and their geometric features. In particular, we show that the moduli space of univariate normal distributions gives rise to a correspondence between the Siegel half-space and the Siegel-Jacobi space, which are spaces that appear in the context of automorphic forms.

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Section: The Moduli Of Abelian Varietiesmentioning

confidence: 82%