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arXiv · 2401.02910

Kähler metrics and toric Lagrangian fibrations

Rui Loja Fernandes·Maarten Mol
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Abstract

We extend the Abreu-Guillemin theory of invariant Kähler metrics from toric symplectic manifolds to any symplectic manifold admitting a toric action of a symplectic torus bundle. We show that these are precisely the symplectic manifolds admitting a Lagrangian fibration with elliptic singularities. The base of such a toric Lagrangian fibration is a codimension 0 submanifold with corners of an integral affine manifold, called a Delzant subspace. This concept generalizes the Delzant polytope associated with a compact symplectic toric manifold. Given a Delzant subspace of finite type, we provide a Delzant-type construction of a Lagrangian fibration with the moment image being the specified Delzant subspace. We establish a 1:1 correspondence between invariant Kähler metrics and a pair consisting of an elliptic connection on the total space of the fibration and a hybrid $b$-metric on the base Delzant subspace, both with specified residues over the facets. Finally, we characterize extremal invariant Kähler metrics as those whose scalar curvature descends to an affine function on the base integral affine manifold. We show that this provides a method for finding and constructing extremal Kähler metrics.

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Rui Loja Fernandes, Maarten Mol. 2026-04-14. Kähler metrics and toric Lagrangian fibrations. https://arxiv.org/abs/2401.02910

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