Vincent Cavalier, Daniel Lehmann, Marcio G. Soares
Classes de Chern Virtuelles des Ensembles Analytiques et Applications
Preprint series: ESI preprints

MSC:

57R20 Characteristic classes and numbers

57R25 Vector fields, frame fields

19E20 Relations with cohomology theories, See also {14Fxx}

Abstract: 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 $V_0$ its regular part, and by $[V]$ its fundamental class
in $H_{2n}(V;\hbox {\bb Z})$.

If $V$ is a locally complete intersection (LCI), it is known that the normal
bundle $N_{V_0}$ in $M$ to $V_0$ in $M$ has a {\it natural} extension $ N_{V } $
to all of $V$, so that we can define its Chern classes $c^{(*)}( N_{V })$, 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 $K^0(V)$. This has applications\hb
- on one hand to the definition of various indices associated to a singular
foliation $\C F$ on $M$ with respect to which $V$ is invariant (cf.
[LS1][LSS][LS2]),\hb
- on the other hand to the definition of the Milnor numbers and
classes of the singular part of $V$ (cf. [BLSS1][BLSS2]).

In the general case, we can no more define $N_{V }$ and $T_{vir}(V)$. However we
shall prove that it is always possible to define Chern classes $c_{n-*}( N_{V } )$
and $c_{n-*}^{vir} (V)$ in the homology $H_{2(n-*)} (V)$, which coincide
respectively with the the Poincar\'{e} duals $ c^{(*)}( N_{V } )\frown [V]$ and $
c_{vir}^* (V) \frown [V]$ of the cohomological Chern classes $c^{(*)}( N_{V })$ and
$c_{vir}^* (V)$ when $V$ is LCI. Moreover, it turns out that this is sufficient for
being able to generalize to all compact pure dimensional complex analytic subsets of
a holomorphic manifold the two kinds of applications mentioned above.
Keywords: coherent sheaf, Chern classes, singular holomorphic foliation, residue, K-theory, Milnor classes, Milnor numbers