Differential Geometry Seminar: Collapsing hyperkähler manifolds

Seminar | October 16 | 1-2 p.m. | 891 Evans Hall

 Valentino Tosatti, Northwestern

 Department of Mathematics

Consider a projective hyperkähler manifolds with a surjective holomorphic map with connected fibers onto a lower-dimensional manifold. In the case the base must be half-dimensional projective space, and the generic fibers are holomorphic Lagrangian tori. I will explain how hyperkähler metrics on the total space with volume of the torus fibers shrinking to zero, collapse smoothly away from the singular fibers to a special Kähler metric on the base, whose metric completion equals the global collapsed Gromov-Hausdorff limit, which has a singular set of real Hausdorff codimension at least 2. The resulting picture is compatible with the Strominger-Yau-Zaslow mirror symmetry, and can be used to prove a conjecture of Kontsevich-Soibelman and Gross-Wilson for large complex structure limits which arise via hyperkähler rotation from this construction. This is joint work with Yuguang Zhang.