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Valentino Tosatti: Lectures

September 12, 2023
TITLE: Holomorphic Lagrangian fibrations and special Kähler geometry

ABSTRACT: Consider a compact hyperkähler manifold (aka irreducible holomorphic symplectic) with a nontrivial fiber space structure onto a lower-dimensional space. Classical work of Matsushita shows that the base must be half-dimensional, and the smooth fibers are holomorphic Lagrangian tori. A basic conjecture is that the base of such holomorphic Lagrangian fibrations should be projective space. I will discuss a new proof, joint with Yang Li, of a theorem of Hwang which shows that this conjecture is true when the base is smooth. Our arguments exploit crucially the differential geometry of a “special Kähler metric” that exists on the base away from the discriminant locus.