Summary
We prove a Riemann–Roch type result for any smooth family of smooth oriented compact manifolds. It describes the class of the conjectural higher determinantal gerbe associated to the fibers of the family.
2000 Mathematics Subject Classifications: 57R20 58J (Primary); 17B65 (Secondary)
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M.F. Atiyah, F. Hirzebruch, Riemann–Roch theorems for differentiable manifolds, Bull. Amer. Math. Soc. 65 (1959), 276–281.
S. Bloch, K 2 and algebraic cycles, Ann. of Math. (2) 99 (1974), 349–379.
R. Bott, On the characteristic classes of groups of diffeomorphisms, L'Enseignment Math. 23 (1977), 209–220.
P. Bressler, R. Nest, B. Tsygan, Riemann–Roch theorems via deformation quantization. I, II, Adv. Math. 167 (2002), no. 1, 1–25, 26–73.
L. Breen, On the classification of 2-gerbes and 2-stacks, Astérisque, 225 (1994).
J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Birkhäuser, Boston, 1993.
J.-L. Brylinski, E. Getzler, The homology of algebras of pseudodifferential symbols and the noncommutative residue, K-Theory 1 (1987), no. 4, 385–403.
K.T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), no. 5, 831–879.
P. Deligne, Le déterminant de la cohomologie, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), 93–177, Contemp. Math., 67, Amer. Math. Soc., Providence, RI, 1987.
P. Deligne, Le groupe fondamental de la droite projective moins trois points, Galois groups over \({\mathbb Q}\) (Berkeley, CA, 1987), 79–297, Math. Sci. Res. Inst. Publ., 16, Springer, New York, 1989.
R. Elkik, Fibrés d'intersections et intégrales de classes de Chern, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 2, 195–226.
B. Feigin, B. Tsygan, Riemann–Roch theorem and Lie algebra cohomology I, Proceedings of the Winter School on Geometry and Physics (Srní, 1988). Rend. Circ. Mat. Palermo (2) Suppl. 21 (1989), 15–52.
D.B. Fuks, Cohomology of Infinite-Dimensional Lie Algebras, Consultants Bureau, New York and London, 1986.
V. Hinich, V. Schechtman, Deformation theory and Lie algebra homology, I,II, Algebra Colloq. 4 (1997), no. 2, 213–240, 291–316.
V. Kac, D. Peterson, Spin and wedge representations of infinite-dimensional Lie algebras and groups, Proc. Natl. Acad. Sci. USA, 78 (1981), 3308–3312.
V. Kac, A. Radul, Quasifinite highest weight modules over the Lie algebra of differential operators on the circle, Comm. Math. Phys. 157 (1993), no. 3, 429–457.
R. Kallstrom, Smooth Modules over Lie Algebroids I, preprint math. AG/9808108.
M. Kapranov, E. Vasserot, Vertex algebras and the formal loop space, Publ. Math. Inst. Hautes Études Sci. 100 (2004), 209–269.
M. Kapranov, E. Vasserot, Formal loops II: a local Riemann–Roch theorem for determinantal gerbes, Ann. Sci. École Norm. Sup. (4) 40 (2007), 113–133.
G. Laumon, L. Moret-Bailly, Champs Algébriques, Springer-Verlag, Berlin, 2000.
J. L. Loday, Cyclic Homology, Second Edition, Springer-Verlag, Berlin, 1998.
J. Lott, Higher-degree analogs of the determinant line bundle, Comm. Math. Phys. 230 (2002), no. 1, 41–69.
K. McKenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213. Cambridge University Press, Cambridge, 2005.
S. Morita, Geometry of Characteristic Classes, Translations of Mathematical Monographs, 199. Iwanami Series in Modern Mathematics, American Mathematical Society, Providence, RI, 2001
R. Nest, B. Tsygan, Algebraic index theorem for families, Adv. Math. 113 (1995), no. 2, 151–205.
A. Pressley, G.B. Segal, Loop Groups, Cambridge University Press, 1986.
A.G. Reiman, M.A. Semenov-Tyan-Shanskii, L.D. Faddeev, Quantum anomalies and cocycles on gauge groups, Funkt. Anal. Appl. 18 (1984), No. 4, 64–72.
J. Tate, Residues of differentials on curves, Ann. Sci. École Norm. Sup. (4) 1 (1968) 149–159.
M. Wodzicki, Cyclic homology of differential operators, Duke Math. J. 54 (1987), no. 2, 641–647.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Bressler, P., Kapranov, M., Tsygan, B., Vasserot, E. (2009). Riemann–Roch for Real Varieties. In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics, vol 269. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4745-2_4
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4745-2_4
Published:
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4744-5
Online ISBN: 978-0-8176-4745-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)