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Residues of complex analytic foliations relative to singular invariant subvarieties. (English) Zbl 0916.32023

Lu, Qi-keng (ed.) et al., Singularities and complex geometry. Seminar on singularities and complex geometry, Beijing, China, June 15–20, 1994. Providence, RI: American Mathematical Society. AMS/IP Stud. Adv. Math. 5, 230-245 (1997).
This article is a summary of the paper [D. Lehmann and T. Suwa, J. Differ. Geom. 42, No. 1, 165-192 (1995; Zbl 0844.32007)]. Central ideas are explained. Let \(W\) be a complex manifold. \({\mathcal F}\) a complex analytic foliation on \(W\) with dimension \(s\) with possibly singularities, and \(V\) a \(p\)-dimensional subvariety of \(W\). We assume the following five conditions: (1) \({\mathcal F}\) is reduced. (2) \(V\) is a local complete intersection. (3) The normal sheaf \(N_C\) of \(V\), which is locally free on \(V\), admits an extension \(N\) as a \(C^\infty\) vector bundle to a neighbourhood \(U\) of \(V\). (4) \(V\) is not contained in the set of singular points of \({\mathcal F}\). (5) \(V\) is left invariant by \({\mathcal F}\).
By \(S({\mathcal F},V)\) we denote the union of the set of singular points of \(V\) and the set of singular points of \({\mathcal F}\) on \(V\). For each compact connected component \(Z\) of \(S({\mathcal F},V)\) and for each sequence of integers \(\Delta=(\delta_1,\delta_2,\dots,\delta_k)\) such that \(1\leq \delta_1 \leq\delta_2 \leq \cdots\leq\delta_k\) and \(d=\sum \delta_i<p-s\), they define an element \(\text{Res}_\Delta({\mathcal F}, V;Z)\in H_{2p-2d} (Z;\mathbb{C})\) called the residue, which depends on the local behavior of \({\mathcal F}\) and \(V\) near \(Z\). They show the equality \[ \sum_Zi_*\text{Res}_\Delta({\mathcal F},V;Z)=j^* \bigl(c_{\delta_1} (N) c_{\delta_2} (N)\cdots c_{\delta_k} (N)\bigr)\cap[V] \] in \(H_{2p-2d}(V; \mathbb{C})\) under the assumption that \(V\) is compact, where \(i:Z \hookrightarrow V\) and \(j:V \hookrightarrow U\) denote the embedding, and \(c_\delta (N)\in H^{2\delta} (U; \mathbb{C})\) denotes the \(\delta\)-th Chern class of \(N\). Moreover, they give an explicit integral formula for the residue when \(s=1\) and the singularity is isolated.
For the entire collection see [Zbl 0899.00029].

MSC:

32S65 Singularities of holomorphic vector fields and foliations
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces

Citations:

Zbl 0844.32007
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