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Leray residue for singular varieties. (English) Zbl 0922.32018
Jakubczyk, Bronisław (ed.) et al., Singularities symposium – Łojasiewicz 70. Papers presented at the symposium on singularities on the occasion of the 70th birthday of Stanisław Łojasiewicz, Cracow, Poland, September 25–29, 1996 and the seminar on singularities and geometry, Warsaw, Poland, September 30–October 4, 1996. Warsaw: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 44, 247-256 (1998).
Let \(K\) be a smooth \(n\)-dimensional hypersurface in a complex manifold \(M\), locally defined by the holomorphic function \(f\) such that \(df\) does not vanish along \(K\). Let \(w\in\Omega^{k+1}(M\setminus K)\) be a closed complex valued \(C^\infty\)-form with the first order pole on \(K\). If \(w={df\over f}\vee r+\Theta\), \(r,\Theta C^\infty\)-forms on \(M\) then the Leray residue form \(\text{Res}(w)\) of \(w\) is defined by \(r| K\).
\(\text{Res}(w)\) is closed and its class in \(H^k(K)\) depends only on the class of \(w\) in \(H^{k+1}(M \setminus K)\). If \(K\) has singularities, the residue form can also be constructed using the so-called division property of the differential \((df \neq 0\) at \(x\in M\) and \(df \vee\alpha=0\) for a smooth \(\ell\)-form \(\alpha\) implies \(\alpha=df \vee\beta\) in a neighbourhood of \(x\) which holds if \(\ell\) is not greater than the co-dimension of the singular locus of \(K)\). Especially in the case of isolated singularities the residue form can be constructed for \(\ell\)-forms with \(\ell\leq n =\dim(K)\).
In the paper, residues of holomorphic \((n + 1)\)-forms are studied mainly in case of isolated singularities, line singularities and normal crossings.
For the entire collection see [Zbl 0906.00013].

32S20 Global theory of complex singularities; cohomological properties
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
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