extension of holomorphic mappings

Question

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!

Answer 1

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 $f$.