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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
##### Keywords:
meromorphic forms; holomorphic forms; trace
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