Leray residue theory and Barlet forms on a singular complete intersection (Q2778340)

This note states the existence, on a local reduced complete intersection \(C\) on a complex manifold \(S\), of the following morphism of sheaves: \(\Omega^q_S(\log C)\overset{\text{res}}\longrightarrow \omega_C^{q-k}\) for any \(q\geq k\). Here \(\Omega^q_S(\log C)\) is the sheaf of multilogarithmic differential forms; on an open set \(U\) where \(C\) is defined by holomorphic functions \(h_1,\dots,h_k\), the multilogarithmic forms \(\omega\) are defined by one of the following equivalent conditions, where \(D_i=\{h_i=0\}\):NEWLINENEWLINENEWLINE(i) \(h_j\omega\in\sum^k_{i=1}\Omega^q_U(*\widehat D_i)\), \(h_j k\omega\in\sum^k_{i=1}\Omega^{q+1}_i(*\widehat D_i)\), \(j=1,\dots,k\),NEWLINENEWLINENEWLINE(ii) \(\exists g\in {\mathcal O}_S(U)\), not identically zero on any component of \(C\), a holomorphic \((q-k)\)-form \(\xi\), a meromorphic \(\eta\in\sum^k_{i=1}\Omega^q_U(*\widehat D_i)\) such that \(g\omega=\frac{dh_1}{h_1}\wedge\dots\wedge \frac{dh_k}{h_k}\wedge\xi+\eta\).NEWLINENEWLINENEWLINEThe morphism Res is explicitly defined on \(U\) by: \(\text{Res}(\omega)=[\xi/g]\).NEWLINENEWLINENEWLINEThe morphism is surjective, so that \(\omega_C^{q-k}\simeq \text{Res }\Omega^q_S(\log C)\), and the kernel is on \(U\) equal to \(\sum^k_{i=1}\Omega^q_U(*\widehat D_i)\).NEWLINENEWLINENEWLINEFor a complete description of the sheaf \(\omega_C^{q-k}\), you can refer to [\textit{D. Barlet}, Lect. Notes Math. 670, 187--204 (1978; Zbl 0398.32009)]. The relation with residues was also studied in [\textit{G. Henkin} and \textit{M. Passare}, Invent. Math. 135, No.~2, 297--328 (1999; Zbl 0932.32012)].