For any $y\in Y\backslash f(S)$ the fiber $X_{y}$ is a smooth $(n-m)$-dimensional complex submanifold of $X$. Since $X$ is Kähler, it follows that $X_{y}$ is a minimal submanifold, and so it has vanishing mean curvature vector. We then use the Nash embedding theorem to isometrically embed $(X,\omega_{X})$ into Euclidean space, and so we have an isometric embedding $X\to\mathbb{R}^{N}$. The length of the mean curvature vector of the composite isometric embedding $X_{y}\to X\to\mathbb{R}^{N}$ is then uniformly bounded independent of $y$, since it depends only on the second fundamental form of $X\to\mathbb{R}^{N}$. Then (3.3) follows from the uniform Sobolev inequality of [A, MS]. Notice that they prove an $L^{1}$ Sobolev inequality, but this implies the stated $L^{2}$ Sobolev inequality thanks to the Hölder inequality. ∎