ScalingStacks

The next step is to prove a Poincaré inequality for the restricted metric $\omega_{y}$. This time the constant will not be uniformly bounded, but it will blow up like a power of $\frac{1}{H}$. To this end, we first estimate the Ricci curvature of $\omega_{y}$. Fix a point $y\in Y\backslash f(S)$ and choose local coordinates $z^{1},\dots,z^{n-m}$ on the fiber $X_{y}$, which extend locally to coordinates in a ball in $X$. Then pick local coordinates $w^{n-m+1},\dots,w^{n}$ near $y\in Y\backslash f(S)$, so that $z^{1},\dots,z^{n-m},z^{n-m+1}=f^{*}(w^{n-m+1}),\dots,z^{n}=f^{*}(w^{n})$ give local holomorphic coordinates on $X$. We can also assume that at the point $y$ the metric $\omega_{Y}$ is the identity. At any fixed point of $X_{y}$ we then have

(3.5)

$$\begin{split}\mathrm{Ric}(\omega_{y})&=-\sqrt{-1}\partial\overline{\partial}\log\frac{\omega_{y}^{n-m}}{dz^{1}\wedge\dots\wedge d\overline{z}^{n-m}}\\ &=-\sqrt{-1}\partial\overline{\partial}\log\frac{\omega_{X}^{n-m}\wedge\omega_{0}^{m}}{dz^{1}\wedge\dots\wedge d\overline{z}^{n}}\\ &=-\sqrt{-1}\partial\overline{\partial}\log H-\sqrt{-1}\partial\overline{\partial}\log\frac{\omega_{X}^{n}}{dz^{1}\wedge\dots\wedge d\overline{z}^{n}}\\ &\geq-\frac{\sqrt{-1}\partial\overline{\partial}H}{H}+\mathrm{Ric}(\omega_{X})|_{X_{y}}\\ &\geq-\left(\frac{C}{H}+C\right)\omega_{y}\geq-\frac{C}{H}\omega_{y},\end{split}$$

where all derivatives are in fiber directions. Combining (3.5) and (2.4) we see that the Ricci curvature of $\omega_{y}$ is bounded below by $-C\sigma^{-\lambda}$. Since the diameter of $\omega_{y}$ is bounded by Lemma 3.3, a theorem of Li-Yau [LY] then shows that the Poincaré constant of $\omega_{y}$ is bounded above by $Ce^{B\sigma^{-\lambda}}$. This proves the following