Elliptic genera of level <i>N</i> for complete intersections

doi:10.1007/s11464-021-0917-6

Front. Math. China

2021, Vol. 16

Issue (4) : 1043-1062 https://doi.org/10.1007/s11464-021-0917-6

RESEARCH ARTICLE

Elliptic genera of level N for complete intersections

Jianbo WANG¹(

), Yuyu WANG², Zhiwang YU¹

¹. School of Mathematics, Tianjin University, Tianjin 300350, China
². College of Mathematical Science, Tianjin Normal University, Tianjin 300387, China

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Abstract

We focus on the elliptic genera of level N at the cusps of a congruence subgroup for any complete intersection. Writing the first Chern class of a complete intersection as a product of an integral coefficient c₁ and a generator of the 2nd integral cohomology group, we mainly discuss the values of the elliptic genera of level N for the complete intersection in the cases of c₁>, =, or<0, In particular, the values about the Todd genus, $A^-genus$ , and A_k-genus can be derived from the elliptic genera of level N.

Keywords Complete intersection, elliptic genera of level N Todd genus, A_k-genus

Corresponding Author(s): Jianbo WANG

Issue Date: 11 October 2021

Cite this article:

Jianbo WANG,Yuyu WANG,Zhiwang YU. Elliptic genera of level N for complete intersections[J]. Front. Math. China, 2021, 16(4): 1043-1062.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-021-0917-6
https://academic.hep.com.cn/fmc/EN/Y2021/V16/I4/1043

1	M Atiyah, F Hirzebruch. Spin-manifolds and group actions. In: Haeiger A, Narasimhan R, eds. Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham). Berlin: Springer, 1970, 18–28 https://doi.org/10.1007/978-3-642-49197-9_3
2	O Benoist. Séparation et propriétéde Deligne-Mumford des champs de modules d'intersections complétes lisse. J Lond Math Soc, 2013, 87(2): 138–156 https://doi.org/10.1112/jlms/jds045
3	R Brooks. The A^−genus of complex hypersurfaces and complete intersections. Proc Amer Math Soc, 1983, 87(3): 528–532
4	B Y Chen. Euler characteristics and codimension of complete intersections. Proc Amer Math Soc, 1978, 71(1): 13–14 https://doi.org/10.1090/S0002-9939-1978-0495826-0
5	A Dessai, M Wiemeler. Complete intersections with S1-action. Transform Groups, 2017, 22(2): 295–320 https://doi.org/10.1007/s00031-017-9418-9
6	J Ewing, S Moolgavkar. Euler characteristics of complete intersections. Proc Amer Math Soc, 1976, 56: 390–391 https://doi.org/10.1090/S0002-9939-1976-0399116-4
7	K È Fel'dman. Hirzebruch genus of a manifold supporting a Hamiltonian circle action. Russian Math Surveys, 2001, 56: 978–979 https://doi.org/10.1070/RM2001v056n05ABEH000446
8	R Herrera. Elliptic genera of level N on complex π2 finite manifolds. C R Math Acad Sci Paris, Ser I, 2007, 344(5): 317–320
9	F Hirzebruch. Elliptic genera of level N for complex manifolds. In: Bleuler K, Werner M, eds. Differential Geometrical Methods in Theoretical Physics. Proceedings of the NATO Advanced Research Workshop and the Sixteenth International Conference held in Como, August 24-29, 1987. NATO Adv Sci Inst Ser C Math Phys Sci, 250. Dordrecht: Kluwer Academic Publishers Group, 1988, 37–63 https://doi.org/10.1007/978-94-015-7809-7_3
10	F Hirzebruch. Topological Methods in Algebraic Geometry. Classics in Math. Berlin: Springer-Verlag, 1995
11	F Hirzebruch, T Berger, R Jung. Manifolds and Modular Forms. Aspects of Math, Vol E20. Braunschweig: Friedr Vieweg and Sohn, 1992 https://doi.org/10.1007/978-3-663-14045-0
12	S Kobayashi. Transformation Groups in Differential Geometry. Classics in Math. Berlin: Springer-Verlag, 1995
13	I M Krichever. Obstructions to the existence of S1-actions. Bordism of ramified coverings. Izv Akad Nauk SSSR Ser Mat, 1976, 40: 828-844 (in Russian); Math USSR-Izv, 1976, 10: 783–797 https://doi.org/10.1070/IM1976v010n04ABEH001814
14	I M Krichever. Generalized elliptic genera and Baker-Akhiezer functions. Mat Zametki, 1990, 47: 34-45 (in Russian); Math Notes, 1990, 47: 132–142 https://doi.org/10.1007/BF01156822
15	A S Libgober. Some properties of the signature of complete intersections. Proc Amer Math Soc, 1980, 79(3): 373–375 https://doi.org/10.1090/S0002-9939-1980-0567975-9
16	A S Libgober, J W Wood. Differentiable structures on complete intersections, I. Topology, 1982, 21(4): 469–482 https://doi.org/10.1016/0040-9383(82)90024-6
17	J B Wang, Z W Yu, Y Y Wang. The Hirzebruch genera of complete intersections, arXiv: 2003.02049
18	E Witten. The index of the Dirac operator in loop space. In: Landweber P S, ed. Elliptic Curves and Modular Forms in Algebraic Topology. Lecture Notes in Math, Vol 1326. Berlin: Springer, 1988, 161–181 https://doi.org/10.1007/BFb0078045

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