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Explore the intricate connections between special Kähler geometry and hyperkähler manifolds in this differential geometry and physics seminar lecture. Delve into the origins of special Kähler geometry within N=2 supersymmetric 4D gauge theories and its significant role in mirror symmetry. Examine the fundamental Donagi-Witten and Freed observation that demonstrates how the base of every algebraic integrable system admits a special Kähler metric while the total space admits a hyperkähler metric. Investigate compact hyperkähler manifolds equipped with algebraic integrable systems, specifically holomorphic Lagrangian torus fibrations, and analyze families of hyperkähler metrics where the volume of torus fibers approaches zero. Understand the collapsing behavior of hyperkähler metrics and their convergence to special Kähler metrics on the base away from the discriminant locus, while discovering insights about the metric completion of the resulting limit spaces.
