GSV-indices as residues (Q2879074)

Let \(U\) be a neighborhood of the origin \(0\in \mathbb C^m\) and \(X\subset U\) a reduced complete intersection with an isolated singularity at \(0\) of pure dimension \(n\geq 1\) defined by a sequence of functions \(f_1,\ldots,f_k\). Let \(B\) be a sufficiently small closed ball around the origin such that \(R = B \setminus X\) has a cone structure over the boundary \(\partial R = L\) (the link of \(X\) at \(0)\) and \(X^\prime\) a neighborhood of \(L\) in \(X.\) Given a \(C^\infty\) non-singular vector field \(v\) on \(X^\prime,\) the residue of \(v\) at \(0\) is defined as \(\text{Res}_{c^n}(s,TU|_{X^\prime},0)\) arising from the localization of the \(n\)-th Chern class \(c^n\) of the holomorphic tangent bundle \(TU\) of \(U\) by the \((k+1)\)-tuple of sections \(s=(v,\partial/\partial f_1|_{X^\prime}, \ldots, \partial/\partial f_k|_{X^\prime})\) of \(TU|_{X^\prime}\); see [\textit{T. Suwa}, Indices of vector fields and residues of singular holomorphic foliations. Paris: Hermann (1998; Zbl 0910.32035)]. NEWLINENEWLINENEWLINENEWLINE Making use of certain integral representation, the author shows that the residue of \(v\) at \(0\) coincides with the Poincaré-Hopf index, the GSV-index [\textit{X. Gómez-Mont} et al., Math. Ann. 291, No. 4, 737--752 (1992; Zbl 0725.32012)] and the virtual index of \(v\), see [\textit{D. Lehmann} et al., Bol. Soc. Bras. Mat., Nova Sér. 26, No. 2, 183--199 (1995; Zbl 0852.32015)]. As an application, an algebraic formula for the GSV-index of holomorphic vector fields on singular plane curves is obtained; it differs from the known expressions from [\textit{X. Gómez-Mont}, J. Algebr. Geom. 7, No. 4, 731--752 (1998; Zbl 0956.32029)]. It should be noted that in standard algebraic-geometry setting the field \(v\) can be considered as the restriction to \(X^\prime\) of some holomorphic vector field \(V\) on \(U\) tangent to \(X\) at its non-singular points and containing \(0\) as a possibly isolated singular point.