Hirzebruch, Friedrich; Höfer, Thomas On the Euler number of an orbifold. (English) Zbl 0679.14006 Math. Ann. 286, No. 1-3, 255-260 (1990). 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 Cited in 2 ReviewsCited in 54 Documents 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) Keywords:action of a finite group; orbifold Euler characteristic; string theory; Hilbert scheme of n points Citations:Zbl 0679.14007; Zbl 0686.14022 PDFBibTeX XMLCite \textit{F. Hirzebruch} and \textit{T. Höfer}, Math. Ann. 286, No. 1--3, 255--260 (1990; Zbl 0679.14006) Full Text: DOI EuDML References: [1] L. Dixon, J. Harvey, C. Vafa, E. Witten: Strings on orbifolds. I. Nucl. Phys.B 261, 678-686 (1985) · doi:10.1016/0550-3213(85)90593-0 [2] L. Dixon, J. Harvey, C. Vafa, E. Witten: Strings on orbifolds. II. Nucl. Phys.B274, 285-314 (1986) · doi:10.1016/0550-3213(86)90287-7 [3] L. Göttsche: Die Betti-Zahlen des Hilbert-Schemas for Unterschemata der Läugen einer glatten Fläche. Diplomarbeit, Bonn 1988 [4] L. Göttsche: The Betti numbers of the Hilbert scheme of points on a smooth projective surface. Math. Ann.286, 193-207 (1990) · Zbl 0679.14007 · doi:10.1007/BF01453572 [5] I.G. Macdonald: The Poincaré polynomial of a symmetric product. Proc. Camb. Phil. Soc.58, 563-568 (1962) · Zbl 0121.39601 · doi:10.1017/S0305004100040573 [6] G. Segal: Letter to F. Hirzebruch, December 1988. (Joint paper with M. Atiyah in preparation) [7] A. Strominger, E. Witten: New manifolds for superstring compactification. Commun. Math. Phys.101, 341-361 (1985) · doi:10.1007/BF01216094 [8] C. Vafa: Strings on orbifolds. In: Links between geometry and mathematical physics, Schloß Ringberg 1987. MPI Preprint 87-58 [9] D. Zagier: Equivariant Pontrjagin classes and application to orbit spaces. (Lecture Notes in Mathematics, Vol. 290). Berlin Heidelberg New York: Springer 1972 · Zbl 0238.57013 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.