A new cohomology theory of orbifold. (English) Zbl 1063.53091

Chen, Weimin; Ruan, Yongbin

Commun. Math. Phys. 248, No. 1, 1-31 (2004).

The authors introduce the orbifold cohomology groups of an almost complex orbifold and the orbifold Dolbeault cohomology groups of a complex orbifold. The main result of the paper is the construction of orbifold cup products on orbifold cohomology groups and orbifold Dolbeault cohomology groups, which make the corresponding total orbifold cohomology into a ring with unity. Their considerations are motivated by the construction of the orbifold quantum cohomology groups.

Reviewer: Mircea Puta (Timişoara)

MSC:

53D45	Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
81T30	String and superstring theories; other extended objects (e.g., branes) in quantum field theory

Keywords:

cohomology; orbifold; cup product

References:

[1]	Baily, Jr., W.: The decomposition theorem for V-manifolds. Am. J. Math. 78, 862–888 (1956) · Zbl 0173.22705 · doi:10.2307/2372472
[2]	Borcea, C.: K3-surfaces with involution and mirror pairs of Calabi-Yau manifolds. In: Mirror Symmetry II. Geene, B., Yau, S.-T.(eds)., Providence, RI: Am. Math. Soc. 2001, pp. 717–743 · Zbl 0939.14021
[3]	Batyrev, V.V., Dais, D.: Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry. Topology 35, 901–929 (1996) · Zbl 0864.14022 · doi:10.1016/0040-9383(95)00051-8
[4]	Bott, R., Tu, L. W.: Differential Forms in Algebraic Topology. GTM 82, 1982 · Zbl 0496.55001
[5]	Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in Analysis, A symposium in honour of Bochner, Princeton, N.J.: Princeton University Press, 1970, pp. 195–199
[6]	Chen, W., Ruan, Y.: Orbifold Gromov-Witten theory. Cont. Math. 310, 25–86 (2002) · Zbl 1091.53058 · doi:10.1090/conm/310/05398
[7]	Dixon, L., Harvey, J., Vafa, C., Witten, E.: Strings on orbifolds. Nucl.Phys. B261, 651 (1985)
[8]	Goresky, M., MacPherson, R.: Intersection homology theory. Topology 19, 135–162 (1980) · Zbl 0448.55004 · doi:10.1016/0040-9383(80)90003-8
[9]	Kawasaki, T.: The signature theorem for V-manifolds. Topology 17, 75–83 (1978) · Zbl 0392.58009 · doi:10.1016/0040-9383(78)90013-7
[10]	Kawasaki, T.: The Riemann-Roch theorem for complex V-manifolds. Osaka J. Math. 16, 151–159 (1979) · Zbl 0405.32010
[11]	Li, An-Min., Ruan, Y.: Symplectic surgery and GW-invariants of Calabi-Yau 3-folds. Invent. Math. 145(1), 151–218 (2001) · Zbl 1062.53073 · doi:10.1007/s002220100146
[12]	Reid, M.: McKay correspondence. Seminarire BOurbaki, Vol. 1999/2000. Asterisque No. 176, 53–72 (2002) · Zbl 0996.14006
[13]	Roan, S.: Orbifold Euler characteristic. Mirror symmetry, II, AMS/IP Stud. Adv. Math. 1, Providence, RI: Am. Math. Soc., , 1997, pp. 129–140
[14]	Ruan, Y.: Surgery, quantum cohomology and birational geometry. Am. Math.Soc.Trans (2), 196, 183–198 (1999) · Zbl 0952.53001
[15]	Satake, I.: The Gauss-Bonnet theorem for V-manifolds. J. Math. Soc. Japan 9, 464–492 (1957) · Zbl 0080.37403 · doi:10.2969/jmsj/00940464
[16]	Scott, P.: The geometries of 3-manifolds. Bull. London. Math. Soc. 15, 401–487 (1983) · Zbl 0561.57001 · doi:10.1112/blms/15.5.401
[17]	Thurston, W.: The Geometry and Topology of Three-Manifolds. Princeton Lecture Notes, 1979
[18]	Voisin, C.: Miroirs et involutions sur les surfaces K3. In: Journées de géométrie algébrique d’Orsay, juillet 92, édité par A. Beauville, O. Debarre, Y. Laszlo, Astérisque 218, 273–323 (1993)
[19]	Zaslow, E.: Topological orbifold models and quantum cohomology rings. Commun. Math. Phys. 156(2), 301–331 (1993) · Zbl 0795.53074 · doi:10.1007/BF02098485