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Jaroslaw Wisniewski: Lectures

June 1, 2020
TITLE: Quaternion-Kähler Manifolds via algebraic torus action on projective contact manifolds

ABSTRACT: Let M be a positive quaternion-Kähler manifold of (real) dimension 4n. A conjecture by LeBrun and Salamon asserts that M is symmetric. The twistor space of M is known to be a projective Fano X manifold of (complex) dimension 2n+1 with reductive group of automorphisms. I will report on two projects which aim at proving LeBrun-Salamon conjecture.

In a joint project with Buczyński and Weber we proved the conjecture for n equal 3 and 4. In a project with Romano, Occhetta, and Sola Conde, we proved the conjecture for quaternion-Kähler manifolds admitting an action of a torus of rank bigger or equal to max(2,(n-3)/2). Both proofs are in complex settings, using action of a Cartan (algebraic) torus in a reductive group on a complex contact manifold. Eventually, via the torus action, the problem is reduced to questions in birational algebraic geometry which apparently are related to classical objects and transformations.

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