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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

An analogue of the Novikov Conjecture in complex algebraic geometry


Author: Jonathan Rosenberg
Journal: Trans. Amer. Math. Soc. 360 (2008), 383-394
MSC (2000): Primary 14E05; Secondary 32Q55, 57R77, 58J20, 58J22, 46L87
Published electronically: June 13, 2007
MathSciNet review: 2342008
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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.


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Additional Information

Jonathan Rosenberg
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: jmr@math.umd.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04320-6
PII: S 0002-9947(07)04320-6
Keywords: Birational invariant, Novikov Conjecture, Todd class, complex projective variety, characteristic number, Mishchenko-Fomenko index
Received by editor(s): February 20, 2006
Published electronically: June 13, 2007
Additional Notes: This work was partially supported by NSF grant number DMS-0504212.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.