HyperKähler Geometry

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.

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.

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.

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.