MAT 614 Topics in Algebraic Geometry: Elements of Hodge theory and hyperkaehler geometry

Fall 2019, TuTh 11:30am-12:50pm, Physics P-122


Instructor: Ljudmila Kamenova
Office: Math Tower 3-115
Office hours: TuTh 10-11:30am 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. Many of the properties of K3 surfaces are very well understood. Some of these properties can be generalized to higher dimensional hyperkaehler manifolds. There have been a number of recent developments in hyperkaehler geometry. In this course we shall introduce the transcendental Hodge lattice, the Mumford-Tate group and hyperkaehler manifolds. The transcendental Hodge lattice of a projective manifold is the smallest Hodge substructure in the p-th cohomology which contains all holomorphic p-forms. We'll discuss Zarhin's classification of transcendental Hodge lattices of K3 type. We'll prove that the direct sum of all transcendental Hodge lattices has a natural algebraic structure, and compute this Hodge algebra explicitly for a hyperkaehler manifold following Verbitsky's results.

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. The Mumford-Tate group.

6. The transcendental Hodge lattice of a projective manifold.

7. Zarhin's classification of transcendental Hodge lattices of K3 type.

8. The Hodge algebra of 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. B. van Geemen, C. Voisin, On a conjecture of Matsushita, IMRN 10 (2016) 3111-3123.

7. Yu. Zarhin, Hodge groups of K3 surfaces, J. Reine Angew. Math. 341 (1983) 193-220.

8. M. Verbitsky, Transcendental Hodge algebra, Sel. Math. New Ser. 23 (2017) 2203-2218.

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