Rigidity and vanishing theorems on ℤ/𝕜 Spin^{𝕔} manifolds

Liu, Bo; Yu, Jianqing

doi:10.1090/S0002-9947-2014-06273-9

Rigidity and vanishing theorems on ${\mathbb {Z}}/k$ Spin$^c$ manifolds
HTML articles powered by AMS MathViewer

by Bo Liu and Jianqing Yu PDF

Trans. Amer. Math. Soc. 367 (2015), 1381-1420 Request permission

Abstract:

In this paper, we first establish an $S^1$-equivariant index theorem for Spin$^c$ Dirac operators on $\mathbb {Z}/k$ manifolds, and then combining this equivariant index theorem with the methods developed by Liu-Ma-Zhang and Taubes, we extend Witten’s rigidity theorem to the case of $\mathbb {Z}/k$ Spin$^c$ manifolds. Among others, our results resolve a conjecture of Devoto.

References

M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. MR 397797, DOI 10.1017/S0305004100049410
M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 1, 71–99. MR 397799, DOI 10.1017/S0305004100052105
Nicole Berline, Ezra Getzler, and Michèle Vergne, Heat kernels and Dirac operators, Grundlehren Text Editions, Springer-Verlag, Berlin, 2004. Corrected reprint of the 1992 original. MR 2273508
Jean-Michel Bismut and Gilles Lebeau, Complex immersions and Quillen metrics, Inst. Hautes Études Sci. Publ. Math. 74 (1991), ii+298 pp. (1992). MR 1188532
Raoul Bott and Clifford Taubes, On the rigidity theorems of Witten, J. Amer. Math. Soc. 2 (1989), no. 1, 137–186. MR 954493, DOI 10.1090/S0894-0347-1989-0954493-5
Xianzhe Dai and Weiping Zhang, Real embeddings and the Atiyah-Patodi-Singer index theorem for Dirac operators, Asian J. Math. 4 (2000), no. 4, 775–794. Loo-Keng Hua: a great mathematician of the twentieth century. MR 1870658, DOI 10.4310/AJM.2000.v4.n4.a4
Jorge A. Devoto, Elliptic genera for $\mathbf Z/k$-manifolds. I, J. London Math. Soc. (2) 54 (1996), no. 2, 387–402. MR 1405063, DOI 10.1112/jlms/54.2.387
Daniel S. Freed and Richard B. Melrose, A mod $k$ index theorem, Invent. Math. 107 (1992), no. 2, 283–299. MR 1144425, DOI 10.1007/BF01231891
Akio Hattori, $\textrm {Spin}^{c}$-structures and $S^{1}$-actions, Invent. Math. 48 (1978), no. 1, 7–31. MR 508087, DOI 10.1007/BF01390060
Akio Hattori and Tomoyoshi Yoshida, Lifting compact group actions in fiber bundles, Japan. J. Math. (N.S.) 2 (1976), no. 1, 13–25. MR 461538, DOI 10.4099/math1924.2.13
Friedrich Hirzebruch, Elliptic genera of level $N$ for complex manifolds, Differential geometrical methods in theoretical physics (Como, 1987) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 250, Kluwer Acad. Publ., Dordrecht, 1988, pp. 37–63. MR 981372
Dale Husemoller, Fibre bundles, 3rd ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994. MR 1249482, DOI 10.1007/978-1-4757-2261-1
Igor Moiseevich Krichever, Generalized elliptic genera and Baker-Akhiezer functions, Mat. Zametki 47 (1990), no. 2, 34–45, 158 (Russian); English transl., Math. Notes 47 (1990), no. 1-2, 132–142. MR 1048541, DOI 10.1007/BF01156822
Peter S. Landweber and Robert E. Stong, Circle actions on Spin manifolds and characteristic numbers, Topology 27 (1988), no. 2, 145–161. MR 948178, DOI 10.1016/0040-9383(88)90034-1
H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
Kefeng Liu, On modular invariance and rigidity theorems, J. Differential Geom. 41 (1995), no. 2, 343–396. MR 1331972
Kefeng Liu, On elliptic genera and theta-functions, Topology 35 (1996), no. 3, 617–640. MR 1396769, DOI 10.1016/0040-9383(95)00042-9
Kefeng Liu and Xiaonan Ma, On family rigidity theorems. I, Duke Math. J. 102 (2000), no. 3, 451–474. MR 1756105, DOI 10.1215/S0012-7094-00-10234-7
Kefeng Liu and Xiaonan Ma, On family rigidity theorems for $\textrm {Spin}^c$ manifolds, Mirror symmetry, IV (Montreal, QC, 2000) AMS/IP Stud. Adv. Math., vol. 33, Amer. Math. Soc., Providence, RI, 2002, pp. 343–360. MR 1969037, DOI 10.1090/amsip/033/23
Kefeng Liu, Xiaonan Ma, and Weiping Zhang, $\textrm {Spin}^c$ manifolds and rigidity theorems in $K$-theory, Asian J. Math. 4 (2000), no. 4, 933–959. Loo-Keng Hua: a great mathematician of the twentieth century. MR 1870666, DOI 10.4310/AJM.2000.v4.n4.a12
Kefeng Liu, Xiaonan Ma, and Weiping Zhang, Rigidity and vanishing theorems in $K$-theory, Comm. Anal. Geom. 11 (2003), no. 1, 121–180. MR 2016198, DOI 10.4310/CAG.2003.v11.n1.a6
Takao Matumoto, Equivariant $K$-theory and Fredholm operators, J. Fac. Sci. Univ. Tokyo Sect. I A Math. 18 (1971), 109–125. MR 0290354
John W. Morgan and Dennis P. Sullivan, The transversality characteristic class and linking cycles in surgery theory, Ann. of Math. (2) 99 (1974), 463–544. MR 350748, DOI 10.2307/1971060
Serge Ochanine, Sur les genres multiplicatifs définis par des intégrales elliptiques, Topology 26 (1987), no. 2, 143–151 (French). MR 895567, DOI 10.1016/0040-9383(87)90055-3
Michael Reed and Barry Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0493420
Clifford Henry Taubes, $S^1$ actions and elliptic genera, Comm. Math. Phys. 122 (1989), no. 3, 455–526. MR 998662
Edward Witten, The index of the Dirac operator in loop space, Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986) Lecture Notes in Math., vol. 1326, Springer, Berlin, 1988, pp. 161–181. MR 970288, DOI 10.1007/BFb0078045
Siye Wu and Weiping Zhang, Equivariant holomorphic Morse inequalities. III. Non-isolated fixed points, Geom. Funct. Anal. 8 (1998), no. 1, 149–178. MR 1601858, DOI 10.1007/s000390050051
Weiping Zhang, Circle actions and $\textbf {Z}/k$-manifolds, C. R. Math. Acad. Sci. Paris 337 (2003), no. 1, 57–60 (English, with English and French summaries). MR 1993996, DOI 10.1016/S1631-073X(03)00279-6