Extension properties of meromorphic mappings with values in non-Kähler complex manifolds.

*(English)*Zbl 1081.32010Denote 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.

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.

Reviewer: Vasily A. Chernecky (Odessa)

##### MSC:

32H04 | Meromorphic mappings in several complex variables |

32D20 | Removable singularities in several complex variables |