# Schedule of the Workshop on Hyperkahler Geometry

## Wednesday, November 30

09:20-10:20 |
Olivier Biquard: On gravitational instantons |

10:20-10:50 |
Coffee |

10:50-12:00 |
Hans-Joachim Hein: On gravitational instantons 2 |

12:00-13:00 |
Mark Haskins: Asymptotically cylindrical Calabi-Yau 3-folds and weak Fano 3-folds |

13:00-15:00 |
Lunch |

15:00-16:00 |
Simon Chiossi: Symplectic forms on 4-manifolds |

16:00-16:30 |
Coffee |

16:30-17:30 |
Uwe Semmelmann: Weitzenboeck formulas for manifolds with special holonomy |

## Thursday, December 1

09:20-10:20 |
Roger Bielawski: Pluricomplex deformations of hyperkaehler manifolds |

10:20-10:50 |
Coffee |

10:50-12:00 |
Vicente Cortés: From cubic polynomials to complete quaternionic Kaehler manifolds |

12:00-13:00 |
Gil Cavalcanti: Generalized Kaehler structures and instantons |

13:00-15:00 |
Lunch |

15:00-16:00 |
Sönke Rollenske: Lagrangian fibrations on hyperkaehler manifolds |

16:00-16:30 |
Coffee |

16:30-17:30 |
Andrew Swann: Multi-moment maps and special holonomy |

18:00 |
Welcome Reception in HIM |

## Friday, December 2

09:20-10:20 |
Phil Boalch: Irregular connections, Dynkin diagrams, and fission |

10:20-10:50 |
Coffee |

10:50-12:00 |
Johannes Nordström: Counting associatives in connected-sum G_{2}-manifolds |

12:00-13:00 |
Andrew Dancer: Hyperkaehler implosion |

13:00-15:00 |
Lunch |

## Abstracts:

Roger Bielawski: Pluricomplex deformations of hyperkaehler manifolds

I'll will describe a new type of geometric structure on complex manifolds. It can be viewed as a deformation of hypercomplex structure, but it also leads to a special type of hypercomplex and hyperkaehler geometry. These structures have both algebro-geometric and differential- geometric descriptions, and there are interesting examples arising from physics. This is joint work with Lorenz Schwachhoefer.

Olivier Biquard: On gravitational instantons

Gravitational instantons are asymptotically locally flat hyperKaehler 4-manifolds. I shall discuss constructions and classification of gravitational instantons, in particular a gluing construction obtained with V. Minerbe.

Phil Boalch: Irregular connections, Dynkin diagrams, and fission

I'll survey some results (both old and new) related to the geometry of moduli spaces of irregular connections on curves. If time permits this will include examples of: 1) new nonlinear geometric braid group actions, 2) new complete hyperkaehler manifolds (including some gravitational instantons) [in work with O. Biquard from 2004], and 3) new ways to glue Riemann surfaces together to obtain (symplectic) generalisations of the complex character varieties of surface groups.

Gil Cavalcanti: Generalized Kaehler structures and instantons

We show how the reduction procedure for generalized Kaehler structures can be used to recover Hitchin's results about the existence of a generalized Kaehler structure on the moduli space of instantons on bundle over a generalized Kaehler manifold. In this setup the proof follows closely the proof of the same claim for the Kaehler case and clarifies some of the stranger considerations from Hitchin's proof.

Simon Chiossi: Symplectic forms on 4-manifolds

The theory of holonomy provides a gratifying framework for understanding systems of sympletic forms on almost Hermitian 4-manifolds. I will describe the general picture and present examples, including the Gibbons-Hawking Ansatz and explicit constructions of almost Kaehler Einstein metrics, in dimension 4 and greater.

Vicente Cortés: From cubic polynomials to complete quaternionic Kaehler manifolds

We explain the supergravity r- and c-maps for mathematicians and show that they preserve completeness. As a consequence, any component H of a hypersurface {h = 1} defined by a homogeneous cubic polynomial h such that -d^{2 }h is a complete Riemannian metric on H defines a complete quaternionic Kaehler manifold of negative scalar curvature. The talk is mainly based on joint work with X. Han and T. Mohaupt to appear in Comm. Math. Phys., see arXiv:1101.5103.

Andrew Dancer: Hyperkaehler implosion

Symplectic implosion is an abelianisation construction in symplectic geometry due to Guillemin-Jeffrey-Sjamaar. In this talk I shall describe joint work with Frances Kirwan and Andrew Swann on developing a hyperkaehler analogue.

Mark Haskins: Asymptotically cylindrical Calabi-Yau 3-folds and weak Fano 3-folds

Since Yau's resolution of the Calabi conjecture we know precisely which smooth compact Kahler manifolds admit Ricci-flat metrics in a fixed Kahler class. In the noncompact (or singular cases) we still have some way to go before we have a complete understanding. In the noncompact case one usually needs to make some further assumptions on the asymptotic geometry to make further headway.

In this talk we describe how results of Tian-Yau together with improvements by Kovalev in many cases reduce the problem of constructing asymptotically cylindrical Calabi-Yau (ACyl CY) manifolds to problems in projective algebraic geometry. For the case of ACyl CY 3-folds we explain how many new topological types of ACyl CY 3-folds can be constructed from so-called weak Fano 3-folds. We describe some of the basic geometry of weak Fano 3-folds, explain how to construct numerous examples, discuss the current state of the classification theory and make comparisons to the classical case of Fano 3-folds.

We sketch why ACyl CY 3-folds are a fundamental ingredient in recent work of Corti, Nordstrom, Pacini and the speaker constructing many new compact G_{2} manifolds, including many containing plenty of special 3-dimensional submanifolds, called associative submanifolds. These applications to G_{2} geometry will be described in detail later in the conference in Nordstrom's talk.

Hans-Joachim Hein: On gravitational instantons 2

This talk will essentially be a sequel to Biquard's lecture: more examples, obtained from a rather different construction, and some observations regarding the moduli space of these examples - for instance, properties of the different algebraic surface structures on the same 4-manifold obtained by hyperkaehler rotation of a preferred one.

Johannes Nordström: Counting associatives in connected-sum G_{2}-manifolds

Associative 3-manifolds are a kind of calibrated submanifolds of manifolds with holonomy G_{2}. The deformations of an associative may be obstructed, but if they are not then the associative is rigid, i.e. the moduli space is discrete. One can therefore attempt to count the number of closed associatives in a homology class and, as part of the higher-dimensional gauge theory programme proposed by Donaldson and Thomas, try to use this to define invariants of G_{2}-manifolds. Such invariants could be applied for instance to distinguish between deformation classes of G_{2} holonomy metrics on the same smooth manifold.

In recent work with Corti, Haskins and Pacini we extend Kovalev's twisted connected sum construction of compact G_{2}-manifolds, and show that the same smooth manifold can arise as the result of many different connected sums. We also construct closed associatives in these G_{2}-manifolds, which are the first examples of associatives in compact G_{2}-manifolds that are known to be rigid. In at least some special cases, all associatives in a homology class arise this way, allowing them to be counted.

Sonke Rollenske: Lagrangian fibrations on hyperkaehler manifolds

I will report on a joint project with Daniel Greb and Christian Lehn investigating the following question of Beauville: if a hyperkaehler manifold contains a complex torus L as a Lagrangian submanifold, does it admit a (meromorphic) Lagrangian fibration with fibre L? I will describe a complete positive answer to Beauville's Question for non-algebraic hyperkaehler manifolds, and give explicit necessary and sufficient conditions for a positive solution in the general case using the deformation theory of the pair (X,L).

Uwe Semmelmann: Weitzenboeck formulas for manifolds with special holonomy

Weitzenboeck formulas are an important tool for linking differential geometry and topology. They may be used for proving the vanishing of Betti numbers under suitable curvature assumptions or for proving the non existence of metrics of positive scalar curvature on manifolds satisfying certain topological conditions. Moreover they are often applied in the proof of eigenvalue estimates for Laplace and Dirac operators. In my talk I will consider Weitzenboeck formulas on Riemannian manifolds with a fixed compact holonomy group, in particular G_{2} and Spin(7). I want to show how to derive all such formulas in a certain recursive procedure. It turns out that finding all possible Weitzenboeck formulas may be reformulated into a problem of linear algebra depending on the representation theory of the holonomy group. In the end the structure of the universal enveloping algebra of the Lie algebra of the holonomy group determines the existence of Weitzenboeck formulas.

Andrew Swann: Multi-moment maps and special holonomy

If a group acts on a manifold preserving a closed differential form then there is a notion of multi-moment map that generalises the usual moment map in symplectic geometry. This talk we given the basic definitions and then show how multi-moment maps may be used to determine the geometry of principal orbits for two-torus actions on manifolds of holonomy G_{2} and three-torus actions on manifolds of holonomy Spin(7) in terms of tri-symplectic geometry of four-manifolds.