Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Koszul complexes, harmonic oscillators, and the Todd class


Author: Jean-Michel Bismut
Journal: J. Amer. Math. Soc. 3 (1990), 159-256
MSC: Primary 58G12; Secondary 32L10, 57R20, 58G05, 58G11
DOI: https://doi.org/10.1090/S0894-0347-1990-1017783-4
MathSciNet review: 1017783
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we construct secondary characteristic classes associated with a short exact sequence of holomorphic Hermitian vector bundles. These secondary invariants are generalized analytic torsion forms associated with a family of elliptic operators. They are calculated in terms of Bott-Chern forms and of a certain complicated characteristic class. In a joint calculation with C. Soulé, we relate this characteristic class to the arithmetic Todd genus of Gillet and Soulé.


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

  • [A] M. F. Atiyah, Circular symmetry and stationary phase approximation, Proc. Conf. in honor of L. Schwartz, Astérisque 131 (1985), 43-59. MR 816738 (87h:58206)
  • [AB] M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. I, Ann. of Math. (2) 86 (1967), 374-407; II, Ann. of Math. (2) 88 (1968), 451-491. MR 0212836 (35:3701)
  • [ABP] M. F. Atiyah, R. Bott, and V. K. Patodi, On the heat equation and the Index Theorem, Invent. Math. 19 (1973), 279-330. MR 0650828 (58:31287)
  • [ABS] M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), 3-38. MR 0167985 (29:5250)
  • [AHS] M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), 425-461. MR 506229 (80d:53023)
  • [BaFM] P. Baum, W. Fulton, and R. Macpherson, Riemannian-Roch for singular varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 101-146. MR 0412190 (54:317)
  • [BeV] N. Berline and M. Vergne, A computation of the equivariant index of the Dirac operator, Bull. Soc. Math. France 113 (1985), 305-345. MR 834043 (87f:58146)
  • [B1] J.-M. Bismut, The Index Theorem for families of Dirac operators: two heat equation proofs, Invent. Math. 83 (1986), 91-151. MR 813584 (87g:58117)
  • [B2] -, Localization formulas, superconnections, and the Index Theorem for families, Comm. Math. Phys. 103 (1986), 127-166. MR 826861 (87f:58147)
  • [B3] -, Equivariant Bott-Chern currents and the Ray-Singer analytic torsion, Math. Ann. (to appear). MR 1060688 (91e:58206)
  • [B4] -, The Witten complex and the degenerate Morse inequalities, J. Differential Geom. 23 (1986), 207-240. MR 852155 (87m:58169)
  • [B5] -, Superconnection currents and complex immersions, Invent. Math. (to appear). MR 1029391 (91b:58240)
  • [B6] -, Transformations différentiables du mouvement Brownien, Proc. Conf. in honor of L. Schwartz, Astérisque 131 (1985), 61-87. MR 816739 (88e:60088)
  • [B7] -, Martingales, the Malliavin calculus and hypoellipticity under general Hörmander's conditions, Z. Wahrsch. Verw. Gebiete 56 (1981), 469-505. MR 621660 (82k:60134)
  • [B8] -, The Atiyah-Singer Theorems: A probabilistic approach. I, J. Funct. Anal. 57 (1984), 56-99. MR 744920 (86g:58128a)
  • [B9] -, Index Theorem and equivariant cohomology on the loop space, Comm. Math. Phys. 98 (1985), 213-237. MR 786574 (86h:58129)
  • [B10] -, Large deviations and the Malliavin calculus, Progress in Math., no. 45, Birkhäuser, Basel, Boston, and Stuttgart, 1984. MR 755001 (86f:58150)
  • [B11] -, The infinitesimal Lefschetz formulas: A heat equation proof, J. Funct. Anal. 62 (1985), 435-457. MR 794778 (87a:58144)
  • [B12] -, Complexe de Koszul, oscillateur harmonique et classe de Todd, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), 111-114. MR 1004951 (90e:58147)
  • [BGS1] J-M. Bismut, H. Gillet, and C. Soulé, Analytic torsion and holomorphic determinant bundles. I, Comm. Math. Phys. 115 (1988), 49-78. MR 929146 (89g:58192a)
  • [BGS2] -, Analytic torsion and holomorphic determinant bundles. II, Comm. Math. Phys. 115 (1988), 79-126. MR 929147 (89g:58192b)
  • [BGS3] -, Analytic torsion and holomorphic determinant bundles. III, Comm. Math. Phys. 115 (1988), 301-351. MR 931666 (89g:58192c)
  • [BGS4] -, Bott-Chern currents and complex immersions, Duke Math. J. (to appear). MR 1047123 (91d:58239)
  • [BGS5] -, Complex immersions and Arakelov geometry, Inst. Hautes Études Sci., preprint M/88/64.
  • [BL1] J.-M. Bismut and G. Lebeau, Immersions complexes et métriques de Quillen, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), 111-114. MR 1004951 (90e:58147)
  • [BL2] -, Complex immersions and Quillen metrics (to appear).
  • [BoC] R. Bott and S. S. Chern, Hermitian vector bundles and the equidistribution of the zeros of their holomorphic sections, Acta Math. 114 (1968), 71-112. MR 0185607 (32:3070)
  • [D] S. Donaldson, Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. 50 (1985), 1-26. MR 765366 (86h:58038)
  • [DeM] C. Dellacherie and P.-A. Meyer, Probabilités et potentiels, Chapitre V à VIII. Théorie des martingales, Hermann, Paris, 1980. MR 566768 (82b:60001)
  • [Ge] E. Getzler, A short proof of the Atiyah-Singer Index Theorem, Topology 25 (1986), 111-117. MR 836727 (87h:58207)
  • [GH] P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York, 1978. MR 507725 (80b:14001)
  • [GS] H. Gillet and C. Soulé, Analytic torsion and the arithmetic Todd genus, Topology (to appear). MR 1081932 (92d:14015)
  • [GlJ] J. Glimm and A. Jaffe, Quantum physics, Springer, Berlin, Heidelberg, and New York, 1987. MR 887102 (89k:81001)
  • [H] N. Hitchin, Harmonic spinors, Adv. in Math. 14 (1974), 1-55. MR 0358873 (50:11332)
  • [IkW] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland, Amsterdam, 1981. MR 1011252 (90m:60069)
  • [IMK] K. Ito and H. McKean, Diffusion processes and their sample paths, Grundl. Math. Wiss., bd. 125, Springer, Berlin, Heidelberg, and New York, 1974. MR 0345224 (49:9963)
  • [IN] K. Ito and M. Nisio, On the convergence of sums of independent Banach space valued random variables, Osaka J. Math. 5 (1968), 35-48. MR 0235593 (38:3897)
  • [KM] F. F. Knudsen and D. Mumford, The projectivity of the moduli space of stable curves, I: Preliminaries on ``det'' and ``div'', Math. Scand. 39 (1976), 19-55. MR 0437541 (55:10465)
  • [Le] P. Levy, Wiener's random functions, and other Laplacian random functions, Proc. Second Berkeley Symposium on Mathematical Statistics and Probability (J. Neyman, ed.), Univ. of California Press, Berkeley, CA, 1951, pp. 171-187. MR 0044774 (13:476b)
  • [Ma] P. Malliavin, Stochastic calculus of variations and hypoelliptic operators, Proc. Conf. on Stochastic Differential Equations, Kyoto, 1976, Wiley, New York, 1978, pp. 195-263. MR 536013 (81f:60083)
  • [MQ] V. Mathai and D. Quillen, Superconnections, Thom classes and equivariant differential forms, Topology 25 (1986), 85-110. MR 836726 (87k:58006)
  • [NO] A. Nikiforov and V. Ouvarov, Eléments de la théorie des fonctions spéciales, Mir, Moscow, 1976. MR 0460738 (57:731)
  • [Q1] D. Quillen, Superconnections and the Chern character, Topology 24 (1985), 89-95. MR 790678 (86m:58010)
  • [Q2] -, Determinants of Cauchy-Riemann operators over a Riemann surface, Funct. Anal. Appl. 14 (1985), 31-34.
  • [RS] D. B. Ray and I. M. Singer, Analytic torsion for complex manifolds, Ann. of Math. (2) 98 (1973), 154-177. MR 0383463 (52:4344)
  • [S] J.-P. Serre, Cours d'arithmétique, Presses Universitaires de France, Paris, 1970. MR 0255476 (41:138)
  • [Si1] B. Simon, Functional integration and quantum physics, Academic Press, New York, 1979. MR 544188 (84m:81066)
  • [Si2] -, Notes on infinite determinants of Hilbert space operators, Adv. in Math. 24 (1977), 244-273. MR 0482328 (58:2401)
  • [W] E. Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), 661-692. MR 683171 (84b:58111)
  • [Y] M. Yor, Remarques sur une formule de P. Lévy, Séminaire de Probabilités, no. XIV, Lecture Notes in Math., vol. 784, Springer, Berlin, Heidelberg, and New York, 1980, pp. 343-346. MR 580140 (82c:60144)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC: 58G12, 32L10, 57R20, 58G05, 58G11

Retrieve articles in all journals with MSC: 58G12, 32L10, 57R20, 58G05, 58G11


Additional Information

DOI: https://doi.org/10.1090/S0894-0347-1990-1017783-4
Keywords: Sheaves and cohomology of sections of holomorphic vector bundles, characteristic classes and numbers
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society