Special Kähler geometry and collapsing

Special Kähler geometry was first discovered in the context of N=2 supersymmetric 4D gauge theories, and it also plays a prominent role in mirror symmetry. A key observation of Donagi-Witten and Freed is that the base of every algebraic integrable system admits a special Kähler metric, while the total space admits a hyperkähler metric. In this talk I will consider compact hyperkähler manifolds with a an algebraic integrable system (i.e. a holomorphic Lagrangian torus fibration), and consider a family of hyperkähler metrics such that the volume of the torus fibers shrinks to zero. I will explain how the hyperkähler metrics must collapse to a special Kähler metric on the base (away from the discriminant locus), and what we can say about the metric completion of the limit.