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On the Euler number of an orbifold. (English) Zbl 0679.14006

For an action of a finite group G on a compact manifold X one can define the ‘orbifold Euler characteristic’ e(X,G) as \(\frac{1}{| G|}\sum e(X^{<g,h>}) \), where summation runs over all commuting pairs (g,h) in G and \(e(X^{<g,h>})\) is the topological Euler characteristic of the common fixed point set. This invariant has been introduced in string theory. We suspect that if X is a complex manifold, G operates trivially on its canonical bundle and X/G has some good resolution of singulariies not affecting the canonical bundle, then e(X,G) equals the topological Euler characteristic of this resolution.
We mention some relations to loop spaces and equivariant K-theory and check our guess in some examples from algebraic geometry. In particular we consider the Hilbert scheme of \(n\quad points\) on an algebraic surface which is a good resolution of the n-th symmetric power of the surface. The Betti numbers of this resolution have been computed by L. Göttsche [“The Betti numbers of the Hilbert scheme of points on a smooth projective surface“, Math. Ann. (to appear; see the following review)].
Reviewer: F.Hirzebruch

MSC:

14F45 Topological properties in algebraic geometry
14C05 Parametrization (Chow and Hilbert schemes)
14H30 Coverings of curves, fundamental group
14L30 Group actions on varieties or schemes (quotients)
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References:

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