Geodesic
Chern Classes of Logarithmic Vector Fields
Journal of Singularities, Tome 5 (2012), pp. 109-114

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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. 
@article{10_5427_jsing_2012_5h,
     author = {Xia Liao},
     title = {Chern {Classes} of {Logarithmic} {Vector} {Fields}},
     journal = {Journal of Singularities},
     pages = {109--114},
     publisher = {mathdoc},
     volume = {5},
     year = {2012},
     doi = {10.5427/jsing.2012.5h},
     url = {http://geodesic.mathdoc.fr/articles/10.5427/jsing.2012.5h/}
}
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Xia Liao. Chern Classes of Logarithmic Vector Fields. Journal of Singularities, Tome 5 (2012), pp. 109-114. doi: 10.5427/jsing.2012.5h

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