An analogue of the Novikov Conjecture in complex algebraic geometry

Rosenberg, Jonathan

doi:10.1090/S0002-9947-07-04320-6

An analogue of the Novikov Conjecture in complex algebraic geometry
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Trans. Amer. Math. Soc. 360 (2008), 383-394 Request permission

Abstract:

We introduce an analogue of the Novikov Conjecture on higher signatures in the context of the algebraic geometry of (nonsingular) complex projective varieties. This conjecture asserts that certain “higher Todd genera” are birational invariants. This implies birational invariance of certain extra combinations of Chern classes (beyond just the classical Todd genus) in the case of varieties with large fundamental group (in the topological sense). We prove the conjecture under the assumption of the “strong Novikov Conjecture” for the fundamental group, which is known to be correct for many groups of geometric interest. We also show that, in a certain sense, our conjecture is best possible.

References

Dan Abramovich, Kalle Karu, Kenji Matsuki, and Jarosław Włodarczyk, Torification and factorization of birational maps, J. Amer. Math. Soc. 15 (2002), no. 3, 531–572. MR 1896232, DOI 10.1090/S0894-0347-02-00396-X
M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 236952, DOI 10.2307/1970717
Paul Baum, William Fulton, and Robert MacPherson, Riemann-Roch and topological $K$ theory for singular varieties, Acta Math. 143 (1979), no. 3-4, 155–192. MR 549773, DOI 10.1007/BF02392091
Jonathan Block and Shmuel Weinberger, Higher Todd classes and holomorphic group actions, Pure Appl. Math. Q. 2 (2006), no. 4, Special Issue: In honor of Robert D. MacPherson., 1237–1253. MR 2282420, DOI 10.4310/PAMQ.2006.v2.n4.a13
Lev Borisov and Anatoly Libgober, Higher elliptic genera, arXiv.org preprint math.AG/0602387, 2006.
Jean-Paul Brasselet, Jörg Schürmann, and Shoji Yokura, Hirzebruch classes and motivic Chern classes for singular spaces, arXiv.org preprint math.AG/0503492, 2005.
James F. Davis, Manifold aspects of the Novikov conjecture, Surveys on surgery theory, Vol. 1, Ann. of Math. Stud., vol. 145, Princeton Univ. Press, Princeton, NJ, 2000, pp. 195–224. MR 1747536
Steven C. Ferry, Andrew Ranicki, and Jonathan Rosenberg (eds.), Novikov conjectures, index theorems and rigidity. Vol. 1, London Mathematical Society Lecture Note Series, vol. 226, Cambridge University Press, Cambridge, 1995. Including papers from the conference held at the Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, September 6–10, 1993. MR 1388294
Steven C. Ferry, Andrew Ranicki, and Jonathan Rosenberg (eds.), Novikov conjectures, index theorems and rigidity. Vol. 2, London Mathematical Society Lecture Note Series, vol. 227, Cambridge University Press, Cambridge, 1995. Including papers from the conference held at the Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, September 6–10, 1993. MR 1388306
Friedrich Hirzebruch, Topological methods in algebraic geometry, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Translated from the German and Appendix One by R. L. E. Schwarzenberger; With a preface to the third English edition by the author and Schwarzenberger; Appendix Two by A. Borel; Reprint of the 1978 edition. MR 1335917
Shigeru Iitaka, Algebraic geometry, North-Holland Mathematical Library, vol. 24, Springer-Verlag, New York-Berlin, 1982. An introduction to birational geometry of algebraic varieties. MR 637060
P. J. Kahn, Characteristic numbers and oriented homotopy type, Topology 3 (1965), 81–95. MR 172306, DOI 10.1016/0040-9383(65)90071-6
Jerome Kaminker and John G. Miller, Homotopy invariance of the analytic index of signature operators over $C^\ast$-algebras, J. Operator Theory 14 (1985), no. 1, 113–127. MR 789380
G. G. Kasparov, Topological invariants of elliptic operators. I. $K$-homology, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 4, 796–838 (Russian); Russian transl., Math. USSR-Izv. 9 (1975), no. 4, 751–792 (1976). MR 0488027
G. G. Kasparov, The operator $K$-functor and extensions of $C^{\ast }$-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 3, 571–636, 719 (Russian). MR 582160
G. G. Kasparov, Equivariant $KK$-theory and the Novikov conjecture, Invent. Math. 91 (1988), no. 1, 147–201. MR 918241, DOI 10.1007/BF01404917
G. G. Kasparov, $K$-theory, group $C^*$-algebras, and higher signatures (conspectus), Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993) London Math. Soc. Lecture Note Ser., vol. 226, Cambridge Univ. Press, Cambridge, 1995, pp. 101–146. MR 1388299, DOI 10.1017/CBO9780511662676.007
Gennadi Kasparov, Novikov’s conjecture on higher signatures: the operator $K$-theory approach, Representation theory of groups and algebras, Contemp. Math., vol. 145, Amer. Math. Soc., Providence, RI, 1993, pp. 79–99. MR 1216182, DOI 10.1090/conm/145/1216182
Gennadi Kasparov and Georges Skandalis, Groups acting properly on “bolic” spaces and the Novikov conjecture, Ann. of Math. (2) 158 (2003), no. 1, 165–206. MR 1998480, DOI 10.4007/annals.2003.158.165
Matthias Kreck and Wolfgang Lück, The Novikov conjecture, Oberwolfach Seminars, vol. 33, Birkhäuser Verlag, Basel, 2005. Geometry and algebra. MR 2117411, DOI 10.1007/b137100
Igor Mineyev and Guoliang Yu, The Baum-Connes conjecture for hyperbolic groups, Invent. Math. 149 (2002), no. 1, 97–122. MR 1914618, DOI 10.1007/s002220200214
A. S. Miščenko and A. T. Fomenko, The index of elliptic operators over $C^{\ast }$-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 4, 831–859, 967 (Russian). MR 548506
David Mumford, Algebraic geometry. I, Grundlehren der Mathematischen Wissenschaften, No. 221, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties. MR 0453732
Raghavan Narasimhan, Several complex variables, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1971. MR 0342725
Jonathan Rosenberg, $C^{\ast }$-algebras, positive scalar curvature, and the Novikov conjecture, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 197–212 (1984). MR 720934
Yu. P. Solovyov and E. V. Troitsky, $C^*$-algebras and elliptic operators in differential topology, Translations of Mathematical Monographs, vol. 192, American Mathematical Society, Providence, RI, 2001. Translated from the 1996 Russian original by Troitsky; Translation edited by A. B. Sossinskii. MR 1787114, DOI 10.1090/mmono/192
Robert E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. Mathematical notes. MR 0248858
Chin-Lung Wang, $K$-equivalence in birational geometry, arXiv.org preprint math.AG/0204160, 2002.
R. O. Wells Jr., Differential analysis on complex manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 65, Springer-Verlag, New York-Berlin, 1980. MR 608414
Jarosław Włodarczyk, Toroidal varieties and the weak factorization theorem, Invent. Math. 154 (2003), no. 2, 223–331. MR 2013783, DOI 10.1007/s00222-003-0305-8
Guoliang Yu, The Novikov conjecture for groups with finite asymptotic dimension, Ann. of Math. (2) 147 (1998), no. 2, 325–355. MR 1626745, DOI 10.2307/121011
Guoliang Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000), no. 1, 201–240. MR 1728880, DOI 10.1007/s002229900032