Hyperkähler Geometry

Abstract: - Benjamin Bakker (University of Illinois at Chicago): "Towards a BBDGGHKP decomposition theorem for nonprojective Calabi–Yau varieties"

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.

- 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.

- Andrew Swann (Aarhus University): "HyperKähler metrics and symmetries"

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.

complex variablesdifferential geometry

Audience: researchers in the topic

( chat )

Comments: The discussion is open at gitter.im/GTACOS-November2020/. The live discussion with the speakers for this series of talks will be held on November 18, see researchseminars.org/talk/tacos/7/

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Geometry and TACoS

Series comments: The idea of “Geometry & TACoS” is to have regular mathematical conferences on specific themes, related, in a broad sense, to Geometry and Topology of (Almost) Complex Structures. Each conference will consist of three blocks:

- A series of four lectures, by four speakers, which will be available on our YouTube channel at the beginning of each period.

- A discussion platform, available on Gitter, will be active for two weeks after the release of the videos. There, the participants will be able to interchange remarks, comments, and open questions concerning the topics of the four lectures, as well as interact with each other, the organizers and with the speakers (both in a public and a private way).

- The organizers will follow the development of the discussion and collect questions for the third part of the event, which is an informal coffee break (or TACoS break?) on Zoom, moderated by the organizers, during which the speakers may report on questions and comments that were asked in the chat rooms, and interact personally with the participants.

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