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Abelian differentials on singular varieties and variations on a theorem of Lie-Griffiths. (English) Zbl 0932.32012
Let \(V\) be a complex analytic variety of pure dimension \(n\) in a domain \(D\) in \(\mathbb{C}^{n+p}\) and let \(W\) be a subvariety containing the singular locus of \(V\) but not containing any irreducible component of \(V\). Let \(\psi\) be a holomorphic \(q\)-form on the manifold \(V\setminus W\).
The authors first call \(\psi\) meromorphic on \(V\) if \(\psi\) is locally the restriction to \(V \setminus W\) of a meromorphic form \(\Psi\) in a neighbourhood of \(V\). After proving the equivalence of some properties for \(\psi\) the authors define the notion of holomorphic form for \(\psi\).The authors next define the notion of the trace \(\text{Tr} \psi\) of \(\psi\) when \(D\subset\mathbb{P}^{n+p}\) (complex projective space) is a \(p\)-concave domain with dual domain \(D'\) in the Grassmannian \(\mathbb{G} (p,n+p)\) and \(V\) is a closed analytic purely \(n\)-dimensional variety in \(D\). The trace \(\text{Tr} \psi\) is a meromorphic form on \(D'\) if \(\psi\) is meromorphic in \(V\). Using the notions of meromorphic (or holomorphic) \(n\)-forms and their continuations the authors prove an analytic continuation theorem of analytic varieties in a \(p\)-concave domain \(D\) in \(\mathbb{P}^{n+p}\).

MSC:
32C30 Integration on analytic sets and spaces, currents
32D15 Continuation of analytic objects in several complex variables
32F10 \(q\)-convexity, \(q\)-concavity
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