On the complex formed by contracting differential forms with a vector field on a hypersurface singularity (Q1598346)

The authors consider a germ \((V,0)\subset(\mathbb C^{n+1},0)\) of an analytic hypersurface with an isolated singularity at 0 and a germ of a holomorphic vector field \(X\) tangent to it. The field \(X\) is supposed to be the restriction to \(V\) of a germ of a holomorphic vector field \(\widetilde X\) on \((\mathbb C^{n+1},0)\). The \textit{contraction with} \(X\) of a holomorphic \(i\)- form \(\omega\) on \(V\) is the \(i-1\)- form \(i_X\omega\) obtained by substituting \(X\) to \(\omega\). The contraction defines a complex of holomorphic \(i\)- forms of \(V\), \(i=0,\dots, n+1\), over the ring of germs of holomorphic functions at 0. (The forms in the complex are taken modulo those divisible by the function (denoted \(f\)) generating the ideal of the hypersurface and those divisible by its differential.) As it was shown in [\textit{X. Gómez-Mont}, J. Algebr. Geom. 7, No. 4, 731-752 (1998; Zbl 0956.32029)], all the homology groups \[ H_i,\quad i=1,\dots,n+1.\tag{1} \] of the previous complex have the same dimension. In the same paper the author considers two modules over the ring of holomorphic functions on \((\mathbb C^{n+1},0)\): the first one (denoted by \(\mathbf A\)) is the quotient of the latter ring by the gradient ideal of the previous function \(f\) defining the hypersurface; the second one (denoted by \(\mathbf B\)) is the quotient of the same ring by the ideal generated by the components of the vector field \(\widetilde X\). Let \[ h=\frac{Df(\widetilde X)}f \] (\(h\) is a holomorphic function, since \(f\) generates the ideal of \(V\)). Then the author considers the annulators of \(f\) and \(h\) in the previous modules and their following quotients: \[ \frac{\text{Ann}_{\mathbf A}(f)}{(h)},\;\frac{\text{Ann}_{\mathbf A}(h)}{(f)},\;\frac{\text{Ann}_{\mathbf B}(f)}{(h)},\;\frac{\text{Ann}_{\mathbf B}(h)}{(f)}.\tag{2} \] In the same paper the author shows that the dimension of the homology groups (1) is equal to that of each quotient in (2). In the paper under review the authors present an algorithm providing an isomorphism between the quotient \(\frac{\text{Ann}_{\mathbf B}(h)}{(f)}\) and the homology group \(H_{2j}\). The problem of computation of the dimensions of homology groups (1) is motivated by the formula for the (homological) index of the vector field (obtained in the previously mentioned paper), which is given by the alternated sum of the previous dimensions (and thus, due to the previous isomorphisms, takes a simpler form in the case of a vector field tangent to a hypersurface).