Given $n,c,V$ there is a fixed $k_{1}$ and integer $N$ with the following effect.

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  Any $X$ in ${\mathcal{K}}(n,c,V)$ can be embedded in a linear subspace of $\mbox{${\mathbb{C}}$}\mbox{${\mathbb{P}}$}^{N}$ by sections of $L^{k_{1}}$.

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  Let $X_{i}$ be a sequence in ${\mathcal{K}}(n,c,V)$ with Gromov-Hausdorff limit $X_{\infty}$. Then $X_{\infty}$ is homeomorphic to a normal projective variety $W$ in $\mbox{${\mathbb{C}}$}\mbox{${\mathbb{P}}$}^{N}$. After passing to a subsequence and taking a suitable sequence of projective transformations we can suppose that the projective varieties $X_{i}\subset\mbox{${\mathbb{C}}$}\mbox{${\mathbb{P}}$}^{N}$ converge as algebraic varieties to $W$.