Calibrated Fibrations on Complete Manifolds via Torus Action

Abstract

In this paper we will investigate torus actions on complete manifolds with calibrations. For Calabi-Yau manifolds M^2n with a Hamiltonian structure-preserving k-torus action we show that any symplectic reduction has a natural holomorphic volume form. Moreover Special Lagrangian (SLag) submanifolds of the reduction lift to SLag submanifolds of M, invariant under the torus action. If k=n-1 and the first cohomology of M is trivial, then we prove that M is a fibration with generic fiber being a SLag submanifold. As an application we will see that crepant resolutions of singularities of a finite Abelian subgroup of SU(n) acting on C^n have SLag fibrations. We study SLag submanifolds on the total space K(N) of a canonical bundle of a Kahler-Einstein manifold N with positive scalar curvature. We give a conjecture about fibration of K(N) by SLag subvarieties with a certain asymptotic behavior at infinity, which we prove if N is toric. We also get similar results for coassociative submanifolds of a G_2-manifold M^7, which admits a 3-torus, a 2-torus or an SO(3)-action.