Chern Classes of Logarithmic Vector Fields

Xia Liao

Journal of Singularities
volume 5 (2012), 109-114
Proceedings of the International Conference on Singularity Theory and Applications, Hefei, China, July 25-31, 2011

Received 9 January 2012. Received in revised form 21 April 2012.

DOI: 10.5427/jsing.2012.5h

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Let X be a nonsingular complex variety and D a reduced effective divisor in X. In this paper we study the conditions under which the formula c_{SM}(1_U)=c(Der_X(-log D))\cap [X] is true. We prove that this formula is equivalent to a Riemann-Roch type of formula. As a corollary, we show that over a surface, the formula is true if and only if the Milnor number equals the Tjurina number at each singularity of D. We also show the Rimann-Roch type of formula is true if the Jacobian scheme of D is nonsingular or a complete intersection.

Author(s) information:

Xia Liao
Mathematics Department
Florida State University
Tallahassee FL 32306, U.S.A.