Proposition 6.9.8B720E73C2654

Proof. Let $p\in Z$ be the given point and $\lambda =\nu (\varphi ,Z)$. Choose $r_0>0$, a constant $C$ and a neighborhood $W$ of $p$ in $Z$ such that for every $y\in W$ the closed ball $\overline{B(y,r_0)}$ lies in the coordinate domain and $\varphi \leq C$ on it. By (6.7.3) at $y$, we have $\varphi (z)\leq \nu (\varphi ,y)\log (|z-y|^2/r_0^2)+C$ for $0<|z-y|<r_0$. Take $z=(z',z'')$ with $y=(0,z'')\in W$ and $0<|z'|<r_0$. Then $|z-y|=|z'|$, the logarithm is negative and $\nu (\varphi ,y)\geq \lambda $, so $\varphi (z)\leq \lambda \log |z'|^2-\lambda \log r_0^2+C$. This is the asserted bound, with a constant independent of $z''$. Conversely, suppose that $\varphi (z)\leq c\log |z'|^2+C'$ for $z'\neq 0$ near a point $q$ of $Z$, where we may assume $c\geq 0$. For $y\in Z$ near $q$ we have $|z'|\leq |z-y|$, so $\varphi (z)\leq c\log |z-y|^2+C'$ for $z\notin Z$ near $y$. Since $Z=\{ z'=0\} $ is a Lebesgue null set in the chart, proposition 6.7.1 gives $\nu (\varphi ,y)\geq c$. By proposition 6.9.7 there are points $y$ arbitrarily close to $q$ with $\nu (\varphi ,y)=\lambda $, so $c\leq \lambda $.

Now let $\pi $ be the blowup of $x$, and let $n=\dim _ xX$. If $n=1$, then $\pi $ is the identity and $E=\{ x\} $, so there is nothing to prove. Let $n\geq 2$ and choose coordinates centered at $x$. The blowup is covered by $n$ charts; in the $k$th chart, $z_ k=w_ k$ and $z_ j=w_ kw_ j$ for $j\neq k$, so that

By (6.7.3), $\varphi \leq \nu (\varphi ,x)\log |z|^2+C$ near $x$. Pulling back, we get $\varphi \circ \pi \leq \nu (\varphi ,x)\log |w_ k|^2+C'$ on compact subsets of the $k$th chart near $E$; since $|w_ k|$ is at most the distance to any point of $E$ in that chart, $\nu (\varphi \circ \pi ,q)\geq \nu (\varphi ,x)$ for every $q\in E$, and $\nu (\varphi ,E)\geq \nu (\varphi ,x)$. Conversely, $E\cong \mathbb P^{n-1}$ is a compact connected submanifold of $Y$, so the first part of the proposition, applied to the weight $\varphi \circ \pi $ and $E$, gives $\varphi \circ \pi \leq \nu (\varphi ,E)\log |w_ k|^2+C$ near each point of $E$, in the coordinates of a chart containing it. Cover $E$ by finitely many such neighborhoods, on each of which $\log |w_ k|^2-\log |z|^2$ is bounded. Since $\pi $ is proper, the inverse image of a small neighborhood of $x$ lies in their union, so $\varphi (z)\leq \nu (\varphi ,E)\log |z|^2+C''$ near $x$, for $z\neq x$. Hence $\nu (\varphi ,E)\leq \nu (\varphi ,x)$. $\square$