Calibrated Fibrations on Complete Manifolds via Torus Action
نویسنده
چکیده
In this paper we will investigate torus actions on complete manifolds with calibrations. For Calabi-Yau manifolds M 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 H(M, R) = 0 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 Cn have SLag fibrations. We study SLag submanifolds on the total space K(N) of the canonical bundle of a Kahler-Einstein manifold N with positive scalar curvature and give a conjecture about fibration of K(N) by SLag sub-varieties, which we prove if N is toric. We will also get similar results for coassociative submanifolds of a G2-manifold M , which admits a 3-torus, a 2-torus or an SO(3) action.
منابع مشابه
Calibrated Fibrations on Noncompact Manifolds via Group Actions
In this paper we use Lie group actions on noncompact Riemannian manifolds with calibrations to construct calibrated submanifolds. In particular, if we have an (n−1)torus acting on a noncompact Calabi-Yau n-fold with a trivial first cohomology, then we have a special Lagrangian fibration on that n-fold. We produce several families of examples for this construction and give some applications to s...
متن کاملLocal Torus Actions Modeled on the Standard Representation
We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants called a characteristic pair and an Euler class of the orbit map, and prove ...
متن کاملHolomorphic curves in Exploded Torus Fibrations: Compactness
The category of exploded torus fibrations is an extension of the category of smooth manifolds in which some adiabatic limits look smooth. (For example, the type of limits considered in tropical geometry appear smooth.) In this paper, we prove a compactness theorem for (pseudo)holomorphic curves in exploded torus fibrations. In the case of smooth manifolds, this is just a version of Gromov’s com...
متن کاملm at h . D G ] 1 3 Se p 20 01 T 3 - fibrations on compact six - manifolds
We describe a simple way of constructing torus fibrations T 3 → X → S which degenerate canonically over a knot or link k ⊂ S. We show that the topological invariants of X can be computed algebraically from the monodromy representation of π1(S 3 \ k) on H1(T ,Z). We use this to obtain some previously unknown T fibrations S×S → S and (S×S)#(S×S)#(S×S) → S whose discriminant locus is a torus knot ...
متن کاملSpecial Lagrangian fibrations. I. Topology,” Integrable systems and algebraic geometry (Kobe/Kyoto
Yau and Zaslow made a surprising conjecture about pairs of mirror manifolds, which, if true, should at last provide a true geometric understanding of mirror symmetry. Simply put, string theory suggests that if X andˇX are mirror pairs of n-dimensional Calabi-Yau manifolds, then on X there should exist a special Lagrangian n-torus fibration f : X → B, (with some singular fibres) such thatˇX is o...
متن کامل