Michel Brion (CNRS Grenoble)
Commutative algebraic groups up to isogeny
The commutative algebraic groups over a field k form an abelian category, which has been studied by Serre and Oort when k is algebraically closed. In particular, they showed that the homological dimension of this category is 1 if k has characteristic 0, and 2 in positive characteristics. Over an arbitrary field k, the homological dimension can be arbitrarily large, as showed by Milne. In this talk, we will consider the isogeny category of commutative algebraic groups, obtained by inverting all morphisms with finite kernel and cokernel. We will see that this category is equivalent to that of modules of finite length over an (explicit but huge) ring, and has homological dimension 1.
Frédéric Campana (Nancy)
Rational curves and negativity of the cotangent bundle
We will survey results relating the existence of covering/connecting families of rational curves on a complex projective manifold to the negativity of its canonical/cotangent bundle. In particular, foliations which are positive in a suitable birational sense have algebraic leaves, with connecting families of rational curves. This extension of former results of Miyaoka and Bogomolov-McQuillan has applications to families of canonically polarised manifolds.
Jun-Muk Hwang (KIAS)
Contact manifold, Legendrian variety and cubic hypersurface
We will discuss three objects: Fano contact manifold, projective Legendrian variety and nondegenerate cubic hypersurface with smooth singular locus. They are related by certain constructions of microlocal nature. We give an overview of these relations and their role in an approach to the LeBrun-Salamon problem on Fano contact manifolds.
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