Projective embeddings and Lagrangian fibrations of Abelian varieties (Q2575646)

The aim of this paper is to construct Lagrangian fibrations on principally polarised abelian varieties as the limit of fibrations induced by the moment map on \({\mathbb P}H^0(L^k)\) for large \(k\), where \(L\) is a symmetric line bundle representing the polarisation. More precisely, given a principally polarised abelian variety \(X\), the author writes down a basis for \(H^0(L^k)\) in terms of theta functions and show that it is orthonormal and balanced with respect to the standard Hermitian and Kähler metrics on \(X\). It follows that the Kähler metric obtained by restricting the Fubini-Study metric on \({\mathbb P}H^0(L^k)\) converges to the standard Kähler metric (with constant scalar curvature) on \(X\): this is deduced from earlier results of Tian, of Zelditch and of Donaldson. By using similar estimates to the ones used for convergence of the metrics, and taking advantage of the action of the Heisenberg group, the author obtains his main result: the moment maps restrict to fibrations on \(X\) which converge, in the Gromov-Hausdorff topology, to a Lagrangian fibration.