June 1, 2020
TITLE: Quaternion-Kähler Manifolds via algebraic torus action on projective contact manifolds
ABSTRACT: Let 
be a positive quaternion-Kähler manifold of (real) dimension 
. A conjecture by LeBrun and Salamon asserts that is symmetric. The twistor space of is known to be a projective Fano X manifold of (complex) dimension 
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 
. 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.
