Assume that $k=\mathbb{C}$ and denote by $n$ the relative dimension of $X$ over $C$. Suppose that $X$ is projective over $C$ and that $K_{X}$ is trivial over $C$. Let $F$ be a general fiber of the morphism $X\to C$. Then $\mathrm{Sk}(X_{K})$ has dimension $n$ if and only if the monodromy transformation around $s\in\mathscr{C}$ on $H^{n}(F(\mathbb{C}),\mathbb{Q})$ has a Jordan block of size $n+1$. If this holds, and $h^{i,0}(F)=0$ for $0<i<n$, then $\mathrm{Sk}(X_{K})$ is a $\mathbb{Q}$-homology sphere.