MAT 614 Topics in Differential Geometry: K3 surfaces and hyperkaehler geometry

Fall 2022, MW 2:40-4:00pm, Earth and Space 181


Instructor: Ljudmila Kamenova

Office: Math Tower 3-115

Office hours: MW 1-2:30pm in Math Tower 3-115; drop by my office anytime or send me an e-mail: kamenova@math.stonybrook.edu.

Hyperkaehler manifolds are (compact) simply connected manifolds with trivial first Chern class that admit an everywhere non-degenerate 2-form. In complex dimension two hyperkaehler surfaces are K3 surfaces. Properties and deformations of K3 surfaces are very well studied. Some of the properties can be generalized to higher dimensional hyperkaehler manifolds. There have been a number of recent developments in hyperkaehler geometry. This course will focus on several main topics connected with them. K3 surfaces provide intuition for studying higher dimensional hyperkaehler manifolds and we shall start by introducing them. We shall construct Mukai moduli spaces and relate them to Hilbert schemes of points on K3 surfaces. Verbitsky proved a global Torelli theorem for the Teichmueller space of compact hyperkaehler manifolds. We'll discuss vanishing of the Kobayashi metric on hyperkaehler manifolds (this is a joint result with S. Lu and M. Verbitsky). We are also going to study twistor spaces and Lagrangian fibrations on hyperkaehler manifolds. Lagrangian fibrations are dense in the moduli space of hyperkaehler manifolds (by a joint result with M. Verbitsky), and thus one can study some of the general properties of hyperkaehler manifolds by studying properties of Lagrangian fibrations.

Tentative Syllabus:

1. K3 surfaces. Kummer surfaces. Local and global Torelli theorems.

2. Basic results and examples of hyperkaehler manifolds.

3. Fibrations and integrable systems. Matsushita-Hwang's theorem.

4. Finiteness results for hyperkaehler manifolds.

5. Veritsky's global Torelli theorem.

6. Ergodicity of complex structures on HK manifolds.

7. Density of Lagrangian fibrations in the HK moduli space.

8. The Kobayashi pseudometric and entire curves on hyperkaehler manifolds.

References:

1. W. Barth, K. Hulek, C. Peters, A. van de Ven, Compact Complex Surfaces, Springer-Verlag, Berlin 2004.

2. D. Huybrechts, Compact hyperkaehler manifolds, Calabi-Yau manifolds and related geometries, Universitext, Springer-Verlag, Berlin, 2003, Lectures from the Summer School held in Nordfjordeid, June 2001, p. 161-225.

3. D. Matsushita, On fibre space structures of a projective irreducible symplectic manifold, Topology 38 (1999), No. 1, 79-83. Addendum, Topology 40 (2001) No. 2, 431-432.

4. J.-M. Hwang, Base manifolds for fibrations of projective irreducible symplectic manifolds, Invent. Math. 174 (2008) No. 3, 625-644.

5. L. Kamenova, Finiteness of Lagrangian fibrations with fixed invariants, C. R. Acad. Sci. Paris 354, Ser. I, No. 7 (2016) 707-711.

6. M. Verbitsky, A global Torelli theorem for hyperkaehler manifolds, Duke Math. J. 162(15) (2013) 2929-2986.

7. M. Verbitsky, Ergodic comples structures on hyperkahler manifolds, Acta Math. 215 (2015) 161-182.

8. L. Kamenova, S. Lu, M. Verbitsky, Kobayashi pseudometric on hyperkaehler manifolds, J. Lond. Math. Soc. (2014) 90(2): 436-450

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