Special Lagrangian submanifolds and Algebraic Complexity one Torus Actions

In the first part of this paper we consider compact algebraic manifolds M 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 H(M ′,R) = 0 then results of [2] show that there is a Special Lagrangian (SLag) fibration on M ′. We will study how the fibers compactify in M and give examples of SLag fibrations on M ′, including some cases there H(M ′,R) 6= 0. In the second part of the paper we study Calabi-Yau hypersurfaces in M . We will 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 ηj near η define smooth Calabi-Yau hypersurfaces Dj in M . We will show that one can deform the SLag fibration on D′ to SLag fibrations on large parts of Dj . This construction applies for instance for Dj being quintics in CP 4 or Calabi-Yau hypersurfaces in the Grassmanian G(2, 4).