extension of holomorphic mappings In 1971 Phillip.A.Griffith proved that
Let $B_n^\ast=\{z\in\mathbb{C}^n:0<\|z\|\le 1\}$ be the punctured ball in $\mathbb{C}^n$,and $f:B_n^\ast\rightarrow M$ a holomorphic mapping into a compact K\"ahler manifold $M$.Then $f$ extends to a meromorphic mapping from the ball $B_n=\{z:\|z\|\le 1\}$ into $M$,provided $n\ge 3$.
We recall the definition of a meromorphic mapping.Let $M$ and $N$ be connected complex manifolds of complex dimension $m$ and $n$ respectively and where $M$ is assumed to be compact.A holomorphic mapping
\begin{equation*}
f:N\rightarrow M
\end{equation*}
which is defined on the complement of a proper subvariety $S\subset N$ will be said to be meromorphic if the closure $\overline{\Gamma}_f$ in $N\times M$ of the graph $\Gamma_f\subset (N-S)\times M$ is an analytic subvariety of $N\times M$.
Griffith also examined the example of the Hopf manifold $M$ (which is not K\"ahler) and showed that $f:B_n^\ast\rightarrow M$ CANNOT extend meromorphically across the origin.
Note that there is a gap between K\"ahler manifolds and the Hopf manifold.Say,balanced manifolds (a balanced manifold $M^m$ is a Hermitian manifold with the K\"ahler form $\omega$ satisfying $d\omega^{m-1}=0$) are not good enough to be K\"ahler but are not as bad as the Hopf manifold.So I want to consider the problem whether $f:B_n^\ast\rightarrow M$ extends meromorphically across the origin when $M$ is a compact balanced manifold.To begin with,I want to check an example when $M$ is an Iwasawa manifold,which is a compact balanced but non-K\"ahler manifold.
Question:
Let $B_n^\ast=\{z\in\mathbb{C}^n:0<\|z\|\le 1\}$ be the punctured ball in $\mathbb{C}^n$,and $f:B_n^\ast\rightarrow M$ a holomorphic mapping into the Iwasawa manifold $M$.Then does $f$ extend to a meromorphic mapping from the ball $B_n=\{z:\|z\|\le 1\}$ into $M$?
Even for this concrete example,I don't know whether the above statement is true.Can you help me to support or reject my viewpoint?Thanks in advance!
 A: First of all, if you are speaking of Phillip Augustus Griffiths, his surname has an "s" at the end. For your specific question about the Iwasawa manifold: the Iwasawa manifold $M$ is the quotient of the complex Heisenberg group $\mathbf{H}_{3,\mathbb{C}}$ by a cocompact, discrete subgroup $\Gamma$.  In particular, the quotient holomorphic map, $$q:\mathbf{H}_{3,\mathbb{C}} \to M,$$ is the universal covering.  
Let $n\geq 1$ be an integer.  The punctured ball $B_n^*$ deformation retracts onto the sphere $\mathbf{S}^{2n-1}$.  If $n\geq 3$, $B_n^*$ is simply connected.  Thus every continuous map $$f: B_n^* \to M,$$ factors through a continuous map to the universal cover, $$\widetilde{f}:B_n^* \to \mathbf{H}_{3,\mathbb{C}}.$$  Since $q$ is a local biholomorphism,  the continuous map $f$ is holomorphic if and only if the continuous map $\widetilde{f}$ is holomorphic.  Finally, as a complex manifold $\mathbf{H}_{3,\mathbb{C}}$ is biholomorphic to $\mathbb{C}^3$.  Therefore, by the extension theorem of Hartogs (sorry for misspelling his name above!), the holomorphic map $\widetilde{f}$ is the restriction to $B_n^*\subset B_n$ of a holomorphic map $$\widetilde{g}:B_n \to \mathbf{H}_{3,\mathbb{C}}.$$  Defining $g = q\circ f$, then $$g:B_n \to M,$$ is a holomorphic extension of $f$.
