Manifolds with holonomy U*(2m)

We consider the geometry determined by a torsion-free affine connection whose holonomy lies in the subgroup U*(2m), a real form of GL(2m,C), otherwise denoted by SL(m,H).U(1). We show in particular how examples may be generated from quaternionic K\"ahler or hyperk\"ahler manifolds with a circle action.Comment: Based on the author's Santalo Lecture delivered in the Universidad Complutense, Madrid on October 10th 2013. To appear in the Revista Matem\'atica Complutens

The moduli space of special Lagrangian submanifolds

This paper considers the natural geometric structure on the moduli space of deformations of a compact special Lagrangian submanifold $L^n$ of a Calabi-Yau manifold. From the work of McLean this is a smooth manifold with a natural $L^2$ metric. It is shown that the metric is induced from a local Lagrangian immersion into the product of cohomology groups $H^1(L)\times H^{n-1}(L)$. Using this approach, an interpretation of the mirror symmetry discussed by Strominger, Yau and Zaslow is given in terms of the classical Legendre transform.Comment: 14 page