I am reading Dominic Joyce's book 'Compact Manifolds with Special Holonomy' and I am struggling to understand a remark he makes at the end of chapter 8. The assertion is (I think) the following:
Suppose $G$ is a finite subgroup of $U(n)$ acting freely on $\mathbb{C}^n\setminus \{0\}$. Let $(X,J,g)$ be a Kähler manifold such that there exists a compact set $K\subset X$ together with a diffeomorphism $\phi:X\setminus K\to \mathbb{C}^n/G\setminus B_R(0)$ satisfying (for $r>R$, and all $k\in \mathbb{N}_{0}$)
1) $|\nabla^k(\phi_{\ast}g-g_0)|_{C^0(\mathbb{C}^n/G\setminus B_r(0))}=O(r^{-2n-k})$
2) $|\nabla^k(\phi_{\ast}J-J_0)|_{C^0(\mathbb{C}^n/G\setminus B_r(0))}=O(r^{-2n-k})$
where $g_0$ is the Euclidean metric on $\mathbb{C}^n/G$ and $J_0$ is the complex structure on $\mathbb{C}^n/G$.
Then:
3) If $n\geq 2$: $(X,J)$ is birational to a deformation of $\mathbb{C}^n/G$.
4) If $n\geq 3$: $(X,J)$ is birational to $\mathbb{C}^n/G$.
I am willing to believe that 4) follows from 3) (At least there is a reference in the book). I would be interested in the proof of 3). Does this follow from a more general theorem in the literature?