Classes de Chern Des Ensembles Analytiques

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• 作者：Vincent Cavalier ; Daniel Lehmann and Marcio G. Soares
• 关键词：Classes de Chern ; faisceau conormal ; feuilletages holomorphes ; r&eacute ; sidus ; K ; th&eacute ; orie ; nombres et classes de Milnor
• 刊名：K-Theory
• 出版年：2005
• 出版时间：January 2005
• 年：2005
• 卷：34
• 期：1
• 页码：35-69
• 全文大小：368 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematicsr>Algebrar>Analysisr>Geometryr>Group Theory and Generalizationsr>
• 出版者：Springer Netherlands
• ISSN：1573-0514

文摘

Let V be a compact complex analytic subset of a non-singular holomorphic manifold M. Assume that V has pure complex dimension n. Denote by V₀ its regular part, and by [V] its fundamental class in H_2n(V;). If V is a locally complete intersection (LCI), it is known that the normal bundle N_{V_0} in M to V₀ in M has a natural extension N_V to all of V, so that we can define its Chern classes c(*)(N_V) in cohomology, as well as the Chern classes c_vir(*). If V is a locally complete intersection (LCI), it is known that the normal bundle N_{V_0} in M to V₀ in M has a natural extension N_V to all of V, so that we can define its Chern classes c(*)(N_V) in cohomology, as well as the Chern classes c_vir(*)(V) of the virtual tangent bundle T_vir(V):=[TM|_V - N_V] in the K-theory K0(V). This has applications– on one hand to the definition of various indices associated to a singular foliationF{\cal F} on M with respect to which V is invariant (cf. [23–25]), and– on the other hand to the definition of the Milnor numbers and classes of the singular part of V (cf. [7,8]).In the general case, we can no more define N_V and T_vir(V). However we shall associate, to each desingularisation cvir_n-*(V, j)c^{\rm vir}_{n-*}(V, \varphi) in the homology H_2(n-*)(V), which coincide respectively with the Poincaré dualsc(*)(N_V)\frown [V]c^{(*)}(N_V)\frown [V] andc(*)_vir(V) \frown [V]c^{(*)}_{\rm vir}(V) \frown [V] of the cohomological Chern classes c(*)(N_V) and c_vir(*)(V) when V is LCI. Our classes do not coincide with the inverse Segre classes and the Fulton–Johnson classes respectively, except for LCIs. Moreover, it turns out that this is sufficient for being able to generalize to compact pure dimensional complex analytic subsets of a holomorphic manifold the two kinds of applications mentioned above. These constructions depend on in general. However, in the case of curves, there is only one desingularisation, so that all these constructions become intrinsic.