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Parity of the spin structure defined by a quadratic differential. (English) Zbl 1064.32010
The spin structure of the moduli space of quadratic differentials on a Riemann surface is studied. It is shown that this spin structure is constant on every stratum where it is defined. This disproves a conjecture that the spin structure classifies the non-hyperelliptic components of the strata of quadratic differentials. The parity of the spin structure is given in explicit form.

MSC:
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F30 Differentials on Riemann surfaces
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
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References:
[1] C Arf, Untersuchungen über quadratische Formen in Körpern der Charakteristik 2 I, J. Reine Angew. Math. 183 (1941) 148 · Zbl 0025.01403
[2] M F Atiyah, Riemann surfaces and spin structures, Ann. Sci. École Norm. Sup. \((4)\) 4 (1971) 47 · Zbl 0212.56402
[3] A Douady, J Hubbard, On the density of Strebel differentials, Invent. Math. 30 (1975) 175 · Zbl 0371.30017
[4] A Eskin, H Masur, A Zorich, Moduli spaces of abelian differentials: the principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math. Inst. Hautes Études Sci. (2003) 61 · Zbl 1037.32013
[5] A Eskin, A Okounkov, Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. Math. 145 (2001) 59 · Zbl 1019.32014
[6] , Travaux de Thurston sur les surfaces, Astérisque 66, Société Mathématique de France (1979) 284 · Zbl 0406.00016
[7] H M Farkas, I Kra, Riemann surfaces, Graduate Texts in Mathematics 71, Springer (1992) · Zbl 0764.30001
[8] R H Fox, R B Kershner, Concerning the transitive properties of geodesics on a rational polyhedron, Duke Math. J. 2 (1936) 147 · Zbl 0014.03401
[9] J Hubbard, H Masur, Quadratic differentials and foliations, Acta Math. 142 (1979) 221 · Zbl 0415.30038
[10] D Johnson, Spin structures and quadratic forms on surfaces, J. London Math. Soc. \((2)\) 22 (1980) 365 · Zbl 0454.57011
[11] M Kontsevich, Lyapunov exponents and Hodge theory, Adv. Ser. Math. Phys. 24, World Sci. Publ., River Edge, NJ (1997) 318 · Zbl 1058.37508
[12] M Kontsevich, A Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003) 631 · Zbl 1087.32010
[13] E Lanneau, Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities, Comment. Math. Helv. 79 (2004) 471 · Zbl 1054.32007
[14] E Lanneau, Connected components of the strata of the moduli spaces of quadratic differentials, Ann. Sci. Éc. Norm. Supér. \((4)\) 41 (2008) 1 · Zbl 1161.30033
[15] H Masur, Interval exchange transformations and measured foliations, Ann. of Math. \((2)\) 115 (1982) 169 · Zbl 0497.28012
[16] H Masur, J Smillie, Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms, Comment. Math. Helv. 68 (1993) 289 · Zbl 0792.30030
[17] J W Milnor, Remarks concerning spin manifolds, Princeton Univ. Press (1965) 55 · Zbl 0132.19602
[18] D Mumford, Theta characteristics of an algebraic curve, Ann. Sci. École Norm. Sup. \((4)\) 4 (1971) 181 · Zbl 0216.05904
[19] G Rauzy, Échanges d’intervalles et transformations induites, Acta Arith. 34 (1979) 315 · Zbl 0414.28018
[20] W A Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. \((2)\) 115 (1982) 201 · Zbl 0486.28014
[21] W A Veech, Moduli spaces of quadratic differentials, J. Analyse Math. 55 (1990) 117 · Zbl 0722.30032
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