Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Holomorphic $ L^2$ torsion without determinant class condition


Author: Guangxiang Su
Journal: Proc. Amer. Math. Soc. 143 (2015), 4513-4524
MSC (2010): Primary 58J52
DOI: https://doi.org/10.1090/S0002-9939-2015-12565-8
Published electronically: March 17, 2015
MathSciNet review: 3373949
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we extend the holomorphic $ L^{2}$ torsion introduced by Carey, Farber and Mathai to the case without the determinant class condition. We compute the metric variation formula for the holomorphic $ L^{2}$ torsion in our case. We also study the asymptotics of the holomorphic $ L^{2}$ torsion associated with a power of a positive line bundle.


References [Enhancements On Off] (What's this?)

  • [1] Jean-Michel Bismut and Éric Vasserot, The asymptotics of the Ray-Singer analytic torsion associated with high powers of a positive line bundle, Comm. Math. Phys. 125 (1989), no. 2, 355-367. MR 1016875 (91c:58141)
  • [2] Maxim Braverman, Alan Carey, Michael Farber, and Varghese Mathai, $ L^2$ torsion without the determinant class condition and extended $ L^2$ cohomology, Commun. Contemp. Math. 7 (2005), no. 4, 421-462. MR 2166660 (2006h:58039), https://doi.org/10.1142/S0219199705001866
  • [3] D. Burghelea, L. Friedlander, T. Kappeler, and P. McDonald, Analytic and Reidemeister torsion for representations in finite type Hilbert modules, Geom. Funct. Anal. 6 (1996), no. 5, 751-859. MR 1415762 (97i:58177), https://doi.org/10.1007/BF02246786
  • [4] A. Carey, M. Farber, and V. Mathai, Determinant lines, von Neumann algebras and $ L^2$ torsion, J. Reine Angew. Math. 484 (1997), 153-181. MR 1437302 (98c:58175)
  • [5] A. Carey, M. Farber, and V. Mathai, Correspondences, von Neumann algebras and holomorphic $ L^2$ torsion, Canad. J. Math. 52 (2000), no. 4, 695-736. MR 1767399 (2001g:58056), https://doi.org/10.4153/CJM-2000-030-7
  • [6] Alan L. Carey and Varghese Mathai, $ L^2$-torsion invariants, J. Funct. Anal. 110 (1992), no. 2, 377-409. MR 1194991 (94a:58211), https://doi.org/10.1016/0022-1236(92)90036-I
  • [7] Michael Farber, von Neumann categories and extended $ L^2$-cohomology, $ K$-Theory 15 (1998), no. 4, 347-405. MR 1656223 (2000b:58041), https://doi.org/10.1023/A:1007778529430
  • [8] John Lott, Heat kernels on covering spaces and topological invariants, J. Differential Geom. 35 (1992), no. 2, 471-510. MR 1158345 (93b:58140)
  • [9] Xiaonan Ma and George Marinescu, Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, vol. 254, Birkhäuser Verlag, Basel, 2007. MR 2339952 (2008g:32030)
  • [10] Varghese Mathai, $ L^2$-analytic torsion, J. Funct. Anal. 107 (1992), no. 2, 369-386. MR 1172031 (93g:58156), https://doi.org/10.1016/0022-1236(92)90114-X
  • [11] Varghese Mathai and Daniel Quillen, Superconnections, Thom classes, and equivariant differential forms, Topology 25 (1986), no. 1, 85-110. MR 836726 (87k:58006), https://doi.org/10.1016/0040-9383(86)90007-8
  • [12] D. Quillen, Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl. 14 (1985), 31-34.
  • [13] Daniel Quillen, Superconnections and the Chern character, Topology 24 (1985), no. 1, 89-95. MR 790678 (86m:58010), https://doi.org/10.1016/0040-9383(85)90047-3
  • [14] D. B. Ray and I. M. Singer, Analytic torsion for complex manifolds, Ann. of Math. (2) 98 (1973), 154-177. MR 0383463 (52 #4344)
  • [15] Mikhail Shubin, De Rham theorem for extended $ L^2$-cohomology, Voronezh Winter Mathematical Schools, Amer. Math. Soc. Transl. Ser. 2, vol. 184, Amer. Math. Soc., Providence, RI, 1998, pp. 217-231. MR 1729936 (2001c:58025)
  • [16] Weiping Zhang, An extended Cheeger-Müller theorem for covering spaces, Topology 44 (2005), no. 6, 1093-1131. MR 2168571 (2006e:58048), https://doi.org/10.1016/j.top.2005.04.001

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 58J52

Retrieve articles in all journals with MSC (2010): 58J52


Additional Information

Guangxiang Su
Affiliation: Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
Email: guangxiangsu@nankai.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2015-12565-8
Keywords: Holomorphic $L^2$ torsion, determinant class condition, positive line bundle
Received by editor(s): June 6, 2014
Received by editor(s) in revised form: June 12, 2014
Published electronically: March 17, 2015
Additional Notes: The author was supported by “the Fundamental Research Funds for the Central Universities 65011541” and NSFC 11101219
Communicated by: Varghese Mathai
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society