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Calculation of Rozansky-Witten invariants on the Hilbert schemes of points on a K3 surface and the generalised kummer varieties. (English) Zbl 1064.53030

A compact hyper-Kähler manifold \((X,g)\) is a compact Riemannian manifold whose holonomy is contained in Sp\((n)\). L. Rozansky and E. Witten [Sel. Math., New Ser. 3, No. 3, 401–458 (1997; Zbl 0908.53027)] describe how one can associate to every vertex-oriented trivalent graph \(\Gamma\) an invariant \(b_\Gamma(X)\) to \(X\). If \((X,g)\) is given the structure of a Kähler manifold then \(X\) carries a holomorpic sympletic two-form and is thus called a holomorphic sympletic manifold. In [Compos. Math. 115, No. 1, 71–113 (1999; Zbl 0993.53026)] M. Kapronov shows how one can calculate \(b_\Gamma(X)\) from \(X\) and the two-form it carries. The two main examples of holomorphic symplectic manifolds are the Hilbert schemes \(X^{[n]}\) of points on a K3 surface \(X\) and the generalized Kummer varieties \(A^{[[n]]}\). Actual calculations for some of these examples are carried out by J. Sawson [“Rozansky-Witten invariants of hyperkähler manifolds”. PhD thesis, University of Cambridge, October (1999); see also “The Rozansky-Witten invariants of hyperkähler manifolds”. Kolár, I. (ed.) et al., Differential geometry and applications. Proceedings of the 7th international conference, DGA 98, and satellite conference of ICM in Berlin, Brno, Czech Republic, August 10–14, 1998. Brno: Masaryk University. 429–436 (1999; Zbl 0946.32014)].
The author proves a conjecture of Sawson for the main examples, for graph homology classes that are in a certain algebra generated by closed polywheels (which may include all of the examples). He can calculate \(b_\Gamma(X)\) without dimension restrictions. The paper includes definitions, examples and the appropriate references to provide an introduction to the subject and the contribution it makes.

MSC:

53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
05C99 Graph theory
53D35 Global theory of symplectic and contact manifolds
53D15 Almost contact and almost symplectic manifolds
32Q15 Kähler manifolds
57R20 Characteristic classes and numbers in differential topology
14Q15 Computational aspects of higher-dimensional varieties
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