Remarks on the abelian convexity theorem

2018/01/05 by Biliotti, Leonardo, Ghigi, Alessandro
#14L24 #Differential Geometry (math.DG) #FOS: Mathematics #Primary 53D20 #Secondary 32M05

paper · doi:10.48550/arxiv.1801.01778

Abstract

This note contains some observations on abelian convexity theorems. Convexity along an orbit is established in a very general setting using Kempf-Ness functions. This is applied to give short proofs of the Atiyah-Guillemin-Sternberg theorem and of abelian convexity for the gradient map in the case of a real analytic submanifold of complex projective space. Finally we give an application to the action on the probability measures.

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