Chern Numbers and Rozansky-Witten Invariants of Compact Hyper-Kahler Manifolds -- Contents -- Preface -- Introduction -- Notation -- 1. Compact hyper-Kahler manifolds and holomorphic symplectic manifolds -- 1.1 Basics on compact hyper-Kahler manifolds -- 1.1.1 Holonomy of Riemannian manifolds -- 1.1.2 Definition of a compact hyper-Kahler manifold -- 1.1.3 Holomorphic symplectic manifolds -- 1.1.4 Deformations of compact complex manifolds -- 1.2 Examples -- 1.2.1 The K3 surface -- 1.2.2 The Hilbert scheme of points on a surface -- 1.2.3 Construction of line bundles and classes in H2 on the Hilbert schemes of points on surfaces -- 1.2.4 Hilbert schemes of points on K3 surfaces -- 1.2.5 Generalised Kummer varieties -- 1.2.6 Further examples -- 1.3 Characteristic classes -- 1.3.1 Symplectic sheaves -- 1.3.2 Characteristic classes of symplectic sheaves -- 1.3.3 Chern numbers of holomorphic symplectic manifolds -- 1.4 The Atiyah class -- 1.4.1 Definition -- 1.4.2 Description in terms of Cech cohomology -- 1.4.3 The Bianchi identity -- 1.4.4 Torsion and the Atiyah class of the tangent bundle -- 1.4.5 The Atiyah class of symplectic sheaves -- 1.4.6 Chern-Weil theory -- 1.5 On the second cohomology group of a hyper-Kahler manifold -- 1.5.1 The period map -- 1.5.2 A vanishing result for polynomials on H2 -- 2. Graph homology -- 2.1 The space of graph homology -- 2.1.1 Jacobi diagrams -- 2.1.2 Chains of Jacobi diagrams -- 2.1.3 Glueing legs and product of Jacobi diagrams -- 2.1.4 Subspaces and ideals -- 2.1.5 The graph homology spaces -- 2.2 Symmetric monoidal categories -- 2.2.1 Definition -- 2.2.2 k-linear categories -- 2.2.3 Global sections -- 2.2.4 External tensor and symmetric algebras -- 2.3 Metric Lie algebra objects -- 2.3.1 Definition -- 2.3.2 Examples from the category of vector spaces.

2.3.3 Morphisms between tensor powers of metric Lie algebra objects -- 2.3.4 The PROP of metric Lie algebras -- 2.3.5 The universality of the PROP of metric Lie algebras -- 2.4 Weight systems -- 2.4.1 Definition -- 2.4.2 Constructions of weight systems -- 2.4.3 Modules of metric Lie algebra objects -- 2.5 Operation with graphs and special graphs -- 2.5.1 Special graphs -- 2.5.2 Operations -- 2.5.3 Closed and connected graphs -- 2.5.4 Polywheels -- 2.5.5 The Hopf algebra structure on the space of graph homology -- 2.6 The Wheeling Theorem -- 2.6.1 The wheeling element -- 2.6.2 Wheeling and the Wheeling Theorem -- 3. Rozansky-Witten theory -- 3.1 The Rozansky-Witten weight system -- 3.1.1 The derived category -- 3.1.2 A metric Lie algebra object in the derived category -- 3.1.3 Rozansky-Witten weight systems -- 3.1.4 Properties of the Rozansky-Witten weight system -- 3.1.5 An inner product on the cohomology of a holomorphic symplectic manifold -- 3.1.6 Rozansky-Witten invariants -- 3.1.7 Complex genera and Rozansky-Witten invariants -- 3.2 Some applications -- 3.2.1 Chebyshev polynomials -- 3.2.2 An application of the Wheeling Theorem -- 3.2.3 On the genus td1/2( )td1/2( ) of an irreducible holomorphic symplectic manifold -- 3.2.4 The L2-norm of the Riemannian curvature tensor of a compact hyper-Kahler manifold -- 3.2.5 The Beauville-Bogomolov form -- 3.2.6 A Hirzebruch-Riemann-Roch formula -- 4. Calculations for the example series -- 4.1 More on the geometry of the Hilbert schemes of points on surfaces -- 4.1.1 The universal family -- 4.1.2 The incidence variety X[n,n+1] -- 4.1.3 Calculations in various K-groups -- 4.1.4 Chern numbers of the Hilbert schemes -- 4.2 Genera of Hilbert schemes of points on surfaces -- 4.2.1 Two decomposition results -- 4.2.2 A structural result on genera of Hilbert schemes of points on surfaces.

4.2.3 Genera of the generalised Kummer varieties -- 4.3 Calculations of the power series A, B, C and D -- 4.3.1 Bott's residue formula -- 4.3.2 How to calculate C and D -- 4.3.3 The calculation of A and B -- 4.3.4 Chern numbers for the example series -- 4.4 Calculations of Rozansky-Witten invariants -- 4.4.1 A lemma from umbral calculus -- 4.4.2 More on Rozansky-Witten invariants of closed graph homology classes -- 4.4.3 A structural result on the Rozansky-Witten weights of closed connected graphs on the example series -- 4.4.4 Explicit calculation -- Bibliography -- Index.