The speakers for the third session, Hyperkähler Geometry, are:
- Daniel Huybrechts (Universität Bonn): “3 families of K3 surfaces”
Abstract. I will review three one-dimensional families of K3 surfaces (twistor, Brauer or Tate-Shafarevich, and Dwork) and explain how, from a purely Hodge-theoretic perspective, they fit into one picture. I am particularly interested in understanding how certain properties propagate along those families.
- Benjamin Bakker (University of Illinois at Chicago): “Towards a BBDGGHKP decomposition theorem for nonprojective Calabi–Yau varieties” – slides
Abstract. Calabi-Yau manifolds are built out of simple pieces by the Beauville–Bogomolov decomposition theorem: any Calabi–Yau Kahler manifold up to an etale cover is a product of complex tori, irreducible holomorphic symplectic manifolds, and strict Calabi-Yau manifolds (which have no holomorphic forms except a holomorphic volume form). Work of Druel–Guenancia–Greb–Horing–Kebekus–Peternell over the last decade has culminated in a generalization of this result to projective Calabi–Yau varieties with the kinds of singularities that arise in the MMP, and the proofs heavily use algebraic methods. In this talk I will describe some work in progress with C. Lehn and H. Guenancia extending the decomposition theorem to nonprojective varieties via deformation theory. I will also discuss applications to the K-trivial case of a conjecture of Peternell asserting that any minimal Kahler space can be approximated by algebraic varieties.
- Andrew Swann (Aarhus University): “HyperKähler metrics and symmetries” – slides
Abstract. HyperKähler metrics are surveyed and discussed from the point of view of Lie group symmetries, so principally in the non-compact case. This includes the Gibbons-Hawking ansatz in dimension four, cotangent bundles, coadjoint orbits. A common theme is quotient constructions and various ideas related to symplectic reduction. Relations to other geometric structures naturally arise and show that metrics of indefinite signature have an important role.
- Claire Voisin (Collège de France): “On the Lefschetz standard conjecture for hyper-Kähler manifolds”
Abstract. The Lefschetz standard conjecture is of major importance in the theory of motives. It is open starting from degree 2 and in that degree, it predicts that any holomorphic 2-form on a smooth projective manifold is induced from a 2-form on a surface by a correspondence. I will discuss some results and further expectations in the hyper-Kähler setting.
The schedule of the conference is the following: