Kahler toric manifolds from dually flat spaces

09/10/2021

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by   Mathieu Molitor, et al.

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We present a correspondence between real analytic Kähler toric manifolds and dually flat spaces, similar to Delzant correspondence in symplectic geometry. This correspondence gives rise to a lifting procedure: if f:M→ M' is an affine isometric map between dually flat spaces and if N and N' are Kähler toric manifolds associated to M and M', respectively, then there is an equivariant Kähler immersion N→ N'. For example, we show that the Veronese and Segre embeddings are lifts of inclusion maps between appropriate statistical manifolds. We also discuss applications to Quantum Mechanics.