# A minimal triangulation of complex projective plane admitting a chess colouring of four-dimensional simplices

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## Abstract

We construct and study a new 15-vertex triangulation *X* of the complex projective plane ℂP^{2}. The automorphism group of *X* is isomorphic to *S* _{4} × *S* _{3}. We prove that the triangulation *X* is the minimal (with respect to the number of vertices) triangulation of ℂP^{2} admitting a chess colouring of four-dimensional simplices. We provide explicit parametrizations for the simplices of *X* and show that the automorphism group of *X* can be realized as a group of isometries of the Fubini-Study metric. We find a 33-vertex subdivision \(
\bar X
\) of the triangulation *X* such that the classical moment mapping *μ*: ℂP^{2} → Δ^{2} is a simplicial mapping of the triangulation \(
\bar X
\) onto the barycentric subdivision of the triangle Δ^{2}. We study the relationship of the triangulation *X* with complex crystallographic groups.

## Keywords

STEKLOV Institute Simplicial Complex Solid Torus Barycentric Subdivision Complex Projective Plane## References

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