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)

 
 

 

A condensed proof of the differential Grothendieck-Riemann-Roch theorem


Author: Man-Ho Ho
Journal: Proc. Amer. Math. Soc. 142 (2014), 1973-1982
MSC (2010): Primary 19K56, 58J20, 19L50, 53C08
DOI: https://doi.org/10.1090/S0002-9939-2014-11948-4
Published electronically: March 12, 2014
MathSciNet review: 3182016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a direct proof that the Freed-Lott differential analytic index is well defined and a condensed proof of the differential Grothendieck-Riemann-Roch theorem. As a byproduct we also obtain a direct proof that the $ \mathbb{R}/\mathbb{Z}$ analytic index is well defined and a condensed proof of the $ \mathbb{R}/\mathbb{Z}$ Grothendieck-Riemann-Roch theorem.


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

  • [1] 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 (2007m:58033)
  • [2] Jean-Michel Bismut, The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs, Invent. Math. 83 (1985), no. 1, 91-151. MR 813584 (87g:58117), https://doi.org/10.1007/BF01388755
  • [3] J. M. Bismut, Eta invariants, differential characters and flat vector bundles, with an appendix by K. Corlette and H. Esnault. Chinese Ann. Math. Ser. B 26 (2005), no. 1, 15-44. MR 2129894 (2006b:58023), https://doi.org/10.1142/S0252959905000038
  • [4] Jean-Michel Bismut and Jeff Cheeger, $ \eta $-invariants and their adiabatic limits, J. Amer. Math. Soc. 2 (1989), no. 1, 33-70. MR 966608 (89k:58269), https://doi.org/10.2307/1990912
  • [5] Ulrich Bunke and Thomas Schick, Smooth $ K$-theory, Astérisque 328 (2009), 45-135 (2010) (English, with English and French summaries). MR 2664467 (2012a:19015)
  • [6] Ulrich Bunke and Thomas Schick, Uniqueness of smooth extensions of generalized cohomology theories, J. Topol. 3 (2010), no. 1, 110-156. MR 2608479 (2011e:55011), https://doi.org/10.1112/jtopol/jtq002
  • [7] Ulrich Bunke and Thomas Schick, Differential K-theory. A survey, Global Differential Geometry (Berlin, Heidelberg) (C. B $ \ddot {\textrm {a}}$r, J. Lohkamp, and M. Schwarz, eds.), Springer Proceedings in Mathematics, vol. 17, Springer-Verlag, 2012, pp. 303-358.
  • [8] Jeff Cheeger and James Simons, Differential characters and geometric invariants, Geometry and topology (College Park, Md., 1983/84) Lecture Notes in Math., vol. 1167, Springer, Berlin, 1985, pp. 50-80. MR 827262 (87g:53059), https://doi.org/10.1007/BFb0075216
  • [9] Xianzhe Dai, Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence, J. Amer. Math. Soc. 4 (1991), no. 2, 265-321. MR 1088332 (92f:58169), https://doi.org/10.2307/2939276
  • [10] Daniel S. Freed and John Lott, An index theorem in differential $ K$-theory, Geom. Topol. 14 (2010), no. 2, 903-966. MR 2602854 (2011h:58036), https://doi.org/10.2140/gt.2010.14.903
  • [11] M.-H. Ho, Remarks on the flat Grothendieck-Riemann-Roch theorem, arXiv:1203.3250v2 (2012), not intended for publication.
  • [12] M. J. Hopkins and I. M. Singer, Quadratic functions in geometry, topology, and M-theory, J. Differential Geom. 70 (2005), no. 3, 329-452. MR 2192936 (2007b:53052)
  • [13] Max Karoubi, $ K$-théorie multiplicative, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), no. 8, 321-324 (French, with English summary). MR 838584 (87d:55005)
  • [14] John Lott, $ {\bf R}/{\bf Z}$ index theory, Comm. Anal. Geom. 2 (1994), no. 2, 279-311. MR 1312690 (95j:58166)
  • [15] James Simons and Dennis Sullivan, Axiomatic characterization of ordinary differential cohomology, J. Topol. 1 (2008), no. 1, 45-56. MR 2365651 (2009e:58035), https://doi.org/10.1112/jtopol/jtm006
  • [16] James Simons and Dennis Sullivan, Structured vector bundles define differential $ K$-theory, Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 2010, pp. 579-599. MR 2732065 (2011k:19009)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 19K56, 58J20, 19L50, 53C08

Retrieve articles in all journals with MSC (2010): 19K56, 58J20, 19L50, 53C08


Additional Information

Man-Ho Ho
Affiliation: Department of Mathematics and Statistics, Boston University, Boston, Massachusetts 02215
Address at time of publication: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
Email: homanho@bu.edu, homanho@hkbu.edu.hk

DOI: https://doi.org/10.1090/S0002-9939-2014-11948-4
Received by editor(s): October 7, 2011
Received by editor(s) in revised form: January 24, 2012, June 18, 2012, and July 17, 2012
Published electronically: March 12, 2014
Dedicated: Dedicated to my father, Kar-Ming Ho
Communicated by: Varghese Mathai
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society