Hopf cyclic cohomology and Hodge theory for proper actions on complex manifolds

doi:10.1007/s11464-018-0727-7

Front. Math. China

2018, Vol. 13

Issue (5) : 1189-1214 https://doi.org/10.1007/s11464-018-0727-7

RESEARCH ARTICLE

Hopf cyclic cohomology and Hodge theory for proper actions on complex manifolds

Xin ZHANG(

)

Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China

Download: PDF(254 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem.

Keywords Cyclic cohomology complex Hodge theory proper action vanishing theorem

Corresponding Author(s): Xin ZHANG

Issue Date: 29 October 2018

Cite this article:

Xin ZHANG. Hopf cyclic cohomology and Hodge theory for proper actions on complex manifolds[J]. Front. Math. China, 2018, 13(5): 1189-1214.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-018-0727-7
https://academic.hep.com.cn/fmc/EN/Y2018/V13/I5/1189

1	Aronszajn N. A unique continuation theorem for solutions of elliptic partial differential equations or inequalities. J Math Pure Appl, 1957, 36: 235–249
2	Connes A, Moscovici H. Cyclic cohomology and Hopf algebra symmetry. Lett Math Phys, 2000, 52: 1–28 https://doi.org/10.1023/A:1007698216597
3	Connes A, Moscovici H. Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry. In: Essays on Geometry and Related Topics, Vol 1. Monogr Enseign Math, 38. Geneva: Enseignement Math, Geneva, 2001, 217–255
4	Crainic M. Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes. Comment Math Helv, 2003, 78: 681–721 https://doi.org/10.1007/s00014-001-0766-9
5	Gorokhovsky A. Secondary characteristic classes and cyclic cohomology of Hopf algebras. Topology, 2002, 41: 993–1016 https://doi.org/10.1016/S0040-9383(01)00015-5
6	Griffiths P, Harris J. Principles of Algebraic Geometry. Pure Appl Math. New York: Wiley-Interscience, 1978
7	Kaminker J, Tang X. Hopf algebroids and secondary characteristic classes. J Noncommut Geom, 2009, 3: 1–25 https://doi.org/10.4171/JNCG/28
8	Khalkhali M, Rangipour B. Para-Hopf algebroids and their cyclic cohomology. Lett Math Phys, 2004, 70: 259–272 https://doi.org/10.1007/s11005-004-4303-6
9	Kowalzig N. Hopf algebroids and their cyclic theory. Ph D Thesis, Utrecht University, 2009.
10	Kowalzig N, Posthuma H. The cyclic theory of Hopf algebroids. J Noncommut Geom, 2011, 5: 423–476 https://doi.org/10.4171/JNCG/82
11	Lu J-H. Hopf algebroids and quantum groupoids. Internat J Math, 1996, 7: 47–70 https://doi.org/10.1142/S0129167X96000050
12	Mathai V, Zhang W. Geometric quantization for proper actions. Adv Math, 2010, 225: 1224–1247 https://doi.org/10.1016/j.aim.2010.03.023
13	Phillips N C. Equivariant K-Theory for Proper Actions. Pitman Res Notes Math Ser, 178. Harlow: Longman Scientific & Technical, 1989
14	Rotman J J. An Introduction to Homological Algebra. New York: Academic Press, 1979
15	Shiffman B, Sommese A J. Vanishing Theorems on Complex Manifolds. Progress in Math, 56. Basel: Birkhäuser, 1985 https://doi.org/10.1007/978-1-4899-6680-3
16	Tang X, Yao Y-J, Zhang W. Hopf cyclic cohomology and Hodge theory for proper actions. J Noncommut Geom, 2013, 7: 885–905 https://doi.org/10.4171/JNCG/138

[1]	Xiaojun CHEN,Hai-Long HER,Shanzhong SUN. Lie bialgebra structure on cyclic cohomology of Fukaya categories[J]. Front. Math. China, 2015, 10(5): 1057-1085.

Viewed

Full text

Abstract

Cited

Shared

Discussed