St. Petersburg Mathematical Journal

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1547-7371(e) ISSN 1061-0022(p)

Riemann-Roch theorem for operations in cohomology of algebraic varieties

Author(s): A. L. Smirnov
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 18 (2006), nomer 5.
Journal: St. Petersburg Math. J. 18 (2007), 837-856.
MSC (2000): Primary 14F25, 14F42, 14F43, 14F99
Posted: August 10, 2007
MathSciNet review: 2301046
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: The Riemann-Roch theorem for multiplicative operations in oriented cohomology theories for algebraic varieties is proved and an explicit formula for the corresponding Todd classes is given. The formula obtained can also be applied in the topological situation, and the theorem can be regarded as a change-of-variables formula for the integration of cohomology classes.

References:

1.: F. Hirzebruch, Topological methods in algebraic geometry, Grundlehren Math. Wiss., Bd. 131, Springer-Verlag New York, Inc., New York, 1966. MR 0202713 (34:2573)
2.: A. Borel and J.-P. Serre, Le théorème de Riemann-Roch, Bull. Soc. Math. France 86 (1958), 97-136. MR 0116022 (22:6817)
3.: M. F. Atiyah and F. Hirzebruch, Riemann-Roch theorems for differentiable manifolds, Bull. Amer. Math. Soc. 65 (1959), 276-281. MR 0110106 (22:989)
4.: E. Dyer, Relations between cohomology theories, Colloquium on Algebraic Topology (August 1-10, 1962), Various Publ. Series, No. 1, Mat. Inst. Aarhus Univ., Aarhus, 1962, pp. 89-93.
5.: A. Suslin and V. Voevodsky, Bloch-Kato conjecture and motivic cohomology with finite coefficients, The Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci., vol. 548, Kluwer Acad. Publ., Dordrecht, 2000, pp. 117-189. MR 1744945 (2001g:14031)
6.: F. Morel and V. Voevodsky, Homotopy category of schemes over a base, Preprint, 1997.
7.: V. Voevodsky, $\mathbf A^1$ -homotopy theory, Doc. Math. 1998, Extra Vol. 1, 579-604. MR 1648048 (99j:14018)
8.: I. Panin and A. Smirnov, Push-forwards in oriented cohomology theories of algebraic varieties, -Theory Preprint Archives no. 459, 2000.
9.: I. Panin, Oriented cohomology theories of algebraic varieties, -Theory 30 (2003), 265-314. MR 2064242 (2005f:14043)
10.: A. L. Smirnov, Orientations and transfers in the cohomology of algebraic varieties, Algebra i Analiz 18 (2006), no. 2, 167-224; English transl., St. Petersburg Math. J. 18 (2007), no. 2, 305-346. MR 2244939 (2007f:14018)
11.: I. Panin, Riemann-Roch theorems for oriented cohomology, Axiomatic, Enriched and Motivic Homotopy Theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 131, Kluwer Acad. Publ., Dordrecht, 2004, pp. 261-333. MR 2061857 (2005g:14025)
12.: H. Miller, Universal Bernoulli numbers and the -transfer, Current Trends in Algebraic Topology. Part 2 (London Ont., 1981), CMS Conf. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 1982, pp. 437-449. MR 0686158 (85b:55029)
13.: M. Hovey, Model categories, Math. Surveys Monogr., vol. 63, Amer. Math. Soc., Providence, RI, 1999. MR 1650134 (99h:55031)
14.: V. Voevodsky, Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Études Sci. No. 98 (2003), 1-57. MR 2031198 (2005b:14038a)
15.: H. Gillet, Riemann-Roch theorems for higher algebraic -theory, Adv. in Math. 40 (1981), 203-289. MR 0624666 (83m:14013)
16.: Ch. Soulé, Opérations en -théorie algébrique, Canad. J. Math. 37 (1985), 488-550. MR 0787114 (87b:18013)
17.: V. M. Bukhshtaber, New methods in cobordism theory, R. Stong. Notes on Cobordism Theory, ``Mir'', Moscow, 1973, Appendix to the Russian ed., pp. 336-365. (Russian) MR 0346789 (49:11513)
18.: M. F. Atiyah and F. Hirzebruch, Cohomologie-Operationen und charakteristische Klassen, Math. Z. 77 (1961), 149-187. MR 0156361 (27:6285)
19.: W. Fulton, Intersection theory, 2nd ed., Ergeb. Math. Grenzgeb. (3), Bd. 2, Springer-Verlag, Berlin, 1998. MR 1644323 (99d:14003)
20.: A. L. Smirnov, Homotopy properties of algebraic vector bundles, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 319 (2004), 261-263; English transl., J. Math. Sci. (N.Y.) 134 (2006), no. 6, 2580-2581. MR 2117860 (2005i:14025)
21.: R. Hartshorne, Algebraic geometry, Grad. Texts in Math., No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157 (57:3116)
22.: R. M. Switzer, Algebraic topology--homotopy and homology, Grundlehren Math. Wiss., Bd. 212, Springer-Verlag, New York-Heidelberg, 1975. MR 0385836 (52:6695)
23.: Yu. B. Rudyak, On Thom spectra, orientability, and cobordism, Springer-Verlag, Berlin, 1998. MR 1627486 (99f:55001)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|