arXiv subjects

Edward Goldstein

Publications and source records attributed to Edward Goldstein.

21 records · Page 2Linked to original sources

arXiv2000

Calibrated Fibrations on Complete Manifolds via Torus Action

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.

math.DG↗

arXiv2000

Minimal Lagrangian tori in Kahler Einstein manifolds

In this paper we use structure preserving torus actions on Kahler-Einstein manifolds to construct minimal Lagrangian submanifolds. Our main result is: Let N^2n be a Kahler-Einstein manifold with positive scalar curvature with an effective T^n-action. Then precisely one regular orbit L of the T-action is a minimal Lagrangian submanifold of N. Moreover there is an (n-1)-torus T^n-1 in T^n and a sequence of non-flat immersed minimal Lagrangian tori L_k in N, invariant under T^n-1 s.t. L_k locally converge to L (in particular the supremum of the sectional curvatures of L_k and the distance between L_k and L go to 0 as k goes to infinity.

math.DG↗

arXiv2000

Special Lagrangian submanifolds and Algebraic complexity one Torus Actions

In the first part of this paper we consider compact algebraic manifolds M^2n with an algebraic (n-1)-Torus action. We show that there is a T-invariant meromorphic section $σ$ of the canonical bundle of M. Any such $σ$ defines a divisor D. On the complement M'=M-D we have a trivialization of the canonical bundle and a T-action. If the first cohomology of M' vanishes, then results of [2] imply that there is a Special Lagrangian (SLag) fibration on M'. We study how the fibers compactify in M and give examples of SLag fibrations on M', including some cases there the first cohomology of M' doesn't vanish. In the second part of the paper we study Calabi-Yau hypersurfaces in M. We assume that $σ$ is an inverse of a holomorphic section $η$ of the anti-canonical bundle of M. We will see that under certain assumptions one can choose a holomorphic volume form on the smooth part D' of the divisor D s.t. orbits of the T-action give a SLag fibration on D' with respect to the metric, induced from M. Transversal sections $η_j$ near $η$ define smooth Calabi-Yau hypersurfaces D_j in M. We will show that one can deform the SLag fibration on D' to SLag fibrations on large parts of D_j. This construction applies for instance for D_j being quintics in CP^4 or Calabi-Yau hypersurfaces in the Grassmanian G(2,4).

math.DG↗