no. 4
A Singularity Theorem for Twistor Spinors
[Un théorème de singularité pour les spineurs-twisteurs]
Belgun, Florin Alexandru1 ; Ginoux, Nicolas2 ; Rademacher, Hans-Bert1
1 Universität Leipzig Mathematisches Institut Johannisgasse 26 04109 Leipzig (Allemagne)
2 Universität Potsdam Institut für Mathematik - Geometrie Am Neuen Palais 10 14469 Potsdam (Allemagne)
Annales de l'Institut Fourier, Tome 57 (2007) no. 4, pp. 1135-1159.

Nous étudions les structures spin sur les orbifolds. Nous montrons en particulier que, si la codimension de l’ensemble des singularités est supérieure à 2, alors une orbifold est spin si et seulement si sa partie lisse l’est. Nous prouvons également que, sur une orbifold compacte, tout spineur-twisteur non identiquement nul admet au plus un zéro qui est alors singulier sauf si l’orbifold est conformément équivalente à une sphère ronde. Nous illustrons l’optimalité de nos résultats sur des exemples.

We study spin structures on orbifolds. In particular, we show that if the singular set has codimension greater than 2, an orbifold is spin if and only if its smooth part is. On compact orbifolds, we show that any non-trivial twistor spinor admits at most one zero which is singular unless the orbifold is conformally equivalent to a round sphere. We show the sharpness of our results through examples.

DOI : 10.5802/aif.2289
Classification : 53C21, 53A30, 32C10
Keywords: Orbifolds, twistor-spinors, ALE spaces
Mot clés : orbifolds, spineurs-twisteurs, espaces ALE
Belgun, Florin Alexandru 1 ; Ginoux, Nicolas 2 ; Rademacher, Hans-Bert 1

1 Universität Leipzig Mathematisches Institut Johannisgasse 26 04109 Leipzig (Allemagne)
2 Universität Potsdam Institut für Mathematik - Geometrie Am Neuen Palais 10 14469 Potsdam (Allemagne) 
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Belgun, Florin Alexandru; Ginoux, Nicolas; Rademacher, Hans-Bert. A Singularity Theorem for Twistor Spinors. Annales de l'Institut Fourier, Tome 57 (2007) no. 4, pp. 1135-1159. doi : 10.5802/aif.2289. http://www.numdam.org/articles/10.5802/aif.2289/

[1] Baum, Helga; Friedrich, Thomas; Grunewald, Ralf; Kath, Ines Twistors and Killing spinors on Riemannian manifolds, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], 124, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1991 | MR | Zbl

[2] Borzellino, J.E. Orbifolds of maximal diameter, Ind. Univ. Math. J., Volume 42 (1993), pp. 37-53 | DOI | MR | Zbl

[3] Borzellino, J.E.; Zhu, S.-H. The splitting theorem for orbifolds, Ill. J. Math., Volume 38 (1994), pp. 679-691 | MR | Zbl

[4] Calabi, E. Métriques kählériennes et fibrés holomorphes, Annal. scient. École Norm. Sup., Volume 12 (1979), pp. 269-294 | Numdam | MR | Zbl

[5] Degeratu, A. Geometrical McKay Correspondence for Isolated Singularities (2003) (math.DG/0302068)

[6] Dong, C.; Liu, K.; Ma, X. On orbifold elliptic genus, Orbifolds in mathematics and physics (Madison, WI, 2001) (Contemp. Math), Volume 310, Amer. Math. Soc., Providence, 2002, pp. 87-105 | MR | Zbl

[7] Eguchi, T.; Hanson, A.J. Asymptotically flat solutions to Euclidean gravity, Phys. Lett., Volume 74B (1978), pp. 249-251

[8] Freedman, D.Z.; Gibbons, G.W.; S.W. Hawking, M. Roček Remarks on supersymmetry and Kähler geometry, Superspace and Supergravity, Cambridge Univ. Press, Proc. Nuffield workshop, Cambridge 1980 (1981)

[9] Friedrich, T. Dirac-Operatoren in der riemannschen Geometrie, Adv. lect. Math., Vieweg Verlag, Braunschweig, 1997 | MR

[10] Gibbons, G.W.; Hawking, S.W. Gravitational multi-instantons, Phys. Lett., Volume 78 B (1978), pp. 430-432

[11] Joyce, D. Compact manifolds with special holonomy, Adv. lect. Math., Oxford Math. Monographs, Oxford, 2000 | MR | Zbl

[12] Kronheimer, P.B. A Torelli-type theorem for gravitational instantons, J. Diff. Geom., Volume 29 (1989), pp. 685-697 | MR | Zbl

[13] Kronheimer, P.B. The construction of ALE spaces as hyperkähler quotients, J. Diff. Geom., Volume 29 (1989), pp. 665-683 | MR | Zbl

[14] Kühnel, W.; Rademacher, H.-B. Twistor Spinors and Gravitational Instantons, Lett. Math. Phys., Volume 38 (1996), pp. 411-419 | DOI | MR | Zbl

[15] Kühnel, W.; Rademacher, H.-B. Conformal completion of U(n)–invariant Ricci flat Kähler metrics at infinity, Zeitschr. Anal. Anwend., Volume 16 (1997), pp. 113-117 | MR | Zbl

[16] Kühnel, W.; Rademacher, H.-B. Asymptotically Euclidean Manifolds and Twistor Spinors, Commun. Math. Phys., Volume 196 (1998), pp. 67-76 | DOI | MR | Zbl

[17] Lichnerowicz, A. Killing spinors, twistor–spinors and Hijazi inequality, J. Geom. Phys., Volume 5 (1988), pp. 2-18 | DOI | MR | Zbl

[18] Petersen, P. Riemannian Geometry, Graduate Texts in Mathematics, 171, Springer, 1998 | MR | Zbl

[19] Sardo-Infirri, A. Partial Resolutions of orbifold singularities via moduli spaces of HYM-type bundles (alg-geom/9610004)

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