Lagrangian fibrations and theta functions

Abstract

It is known that holomorphic sections of an ample line bundle $L$ (and its tensor power $L^k$) on an Abelian variety $A$ are given by theta functions. Moreover, a natural basis of the space of holomorphic sections is related to a certain Lagrangian fibration of $A$. We study projective embeddings of $A$ given by the basis for $L^k$, and show that moment maps of toric actions on the ambient projective spaces, restricted to $A$, approximate the Lagrangian fibration of $A$ for large $k$. The case of Kummer variety is also discussed.