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arXiv · 2608.08918

Toric real loci via moment polytopes

João Camarneiro·Ana Cannas da Silva
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Abstract

Toric real loci are distinguished lagrangian submanifolds of toric symplectic manifolds. We develop a polyhedral model for toric real loci, called a kaleidoscope, based on the restriction of the moment map. This construction provides a direct geometric description of the topology of toric real loci and leads to simple criteria for orientability, together with transparent formulas for the Euler characteristic. We illustrate the theory with numerous examples and count, in every complex dimension up to 9, the smooth toric Fano varieties whose real loci are orientable. The kaleidoscope construction offers a simple, visual, and effective framework for understanding the topology of toric real loci through the combinatorics of their moment polytopes.

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João Camarneiro, Ana Cannas da Silva. 2026-08-09. Toric real loci via moment polytopes. https://arxiv.org/abs/2608.08918

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