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Extension properties of meromorphic mappings with values in non-Kähler complex manifolds. (English) Zbl 1081.32010
Denote by $$\Delta(r)$$ the disk of radius $$r$$ in $$\mathbb C$$, $$\Delta := \Delta(1)$$, and for $$0 < r < 1$$ denote by $$\Delta(r,1) := A\setminus{\overline \Delta}(r)$$ an annulus in $$\mathbb C$$. Let $$\Delta^n(r)$$ denote the polydisk of radius $$r$$ in $$\mathbb C^n$$ and $$\Delta^n := \Delta^n(1)$$. Let $$X$$ be a compact complex manifold and consider a meromorphic mapping $$f$$ from the ring domain $$\Delta^n\times\Delta(r,1)$$ into $$X$$.
The author studies the following: Suppose we know that for some nonempty open subset $$U\subset\Delta^n$$ the map $$f$$ extends onto $$U\times\Delta$$. What is the maximal $$\hat U\supset U$$ such that $$f$$ extends meromorphically onto $$\widehat U\times\Delta$$? This is the so-called Hartogs-type extension problem. If $$\widehat U=\Delta^n$$ for any $$f$$ with values in our $$X$$ and any initial (nonempty!) $$U$$ then one says that the Hartogs-type extension theorem holds for meromorphic mappings into this $$X$$.
The goal of this paper is to initiate the systematic study of extension properties of meromorphic mappings with values in non-Kähler complex manifolds. Let $$h$$ be some Hermitian metric on a complex manifold $$X$$ and let $$\omega_h$$ be the associated (1,1)-form. We call $$\omega_h$$ (and $$h$$ itself) pluriclosed or $$dd^c$$-closed if $$dd^c\omega_h = 0$$. Let $$A$$ be a subset of $$\Delta^{n+1}$$ of Hausdorff $$(2n-1)$$-dimensional measure zero. Take a point $$a\in A$$ and a complex two-dimensional plane $$P\ni a$$ such that $$P\cap A$$ is of zero length. A sphere $$\mathbb S^3 =\{x\in P : \| x-a\| =\varepsilon\}$$ with $$\varepsilon$$ small is called a “transversal sphere” if in addition $$\mathbb S^3\cap A = \varnothing$$. Take a nonempty open $$U\subset\Delta^n$$ and set $$H_U^{n+1}(r) =\Delta^n\times A(r, 1)\cup U\times\Delta$$.
The main result of the paper is the following theorem: Let $$f : H_U^{n+1}(r)\rightarrow X$$ be a meromorphic map into a compact complex manifold $$X$$, which admits a Hermitian metric $$h$$, such that the associated (1,1)-form $$\omega_h$$ is $$dd^c$$-closed. Then $$f$$ extends to a meromorphic map $$\widehat f : \Delta^{n+1}\setminus A\rightarrow X$$, where $$A$$ is a complete $$(n-1)$$-polar, closed subset of $$\Delta^{n+1}$$ of Hausdorff $$(2n-1)$$-dimensional measure zero.
Moreover, if $$A$$ is the minimal closed subset such that $$f$$ extends onto $$\Delta^{n+1}\setminus A$$ and $$A\neq\varnothing$$, then for every transversal sphere $$\mathbb S^3\subset\Delta^{n+1} \setminus A$$, its image $$f(\mathbb S^3)$$ is not homologous to zero in $$X$$. All compact complex surfaces admit pluriclosed Hermitian metric forms.
The author also proves the Hartogs-type extension result for mappings into (reduced, normal) complex spaces with $$dd^c$$-negative metric forms. A number of examples which are useful for the understanding of the extension properties of meromorphic mappings into non-Kähler manifolds are given.

##### MSC:
 32H04 Meromorphic mappings in several complex variables 32D20 Removable singularities in several complex variables
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