10.22 Radial densities and metric completeness623ED03FC0EE

Proof. Define $m$ by the first equation in (10.22.3). It is positive for $s>0$ and tends to zero at the origin. By (10.22.2) and the prescribed outer density, $m(s)=\lambda s/(1+s)$ for large $s$: the two expressions for $m^ n$ have the same limit at infinity and the same derivative for large $s$. Integrate $U'=m/s$, choosing the additive constant so that $U(s)=\lambda \log (1+s)$ for large $s$. The Hessian of $U(|y|^2)$ has eigenvalues $m/s$ and $m'$, and its determinant is $(m/s)^{n-1}m'=D$. Thus $q=U(|y|^2)-\lambda \log (1+|y|^2)$ extends by zero outside the affine chart and defines a positive current that is smooth off the origin. Subtracting its supremum gives the normalization.

Since $U$ is increasing, it is bounded above near zero, so lemma 5.24.7 gives a plurisubharmonic extension across the origin, possibly with the value $-\infty $ there. If $U$ is unbounded, then $m(s)\to 0$ implies $U(s)=o(|\log s|)$: for each $\epsilon >0$, bound $m$ by $\epsilon $ on a small interval and integrate $m/s$. The closed non-pluripolar product equals $\mu _ D$ off the origin and puts no mass at the origin; the measure $\mu _ D$ has no atom there and has mass $\lambda ^ n$ by (10.22.2). Hence $q$ is a full-mass solution, and it is unique by the uniqueness of full-mass solutions in the closed theory, discussed in section 25.21. Integrating $U'=m/s$ shows that $U$ is bounded near the origin precisely under (10.22.4).

Write $R=|y|$. For a path approaching the origin, the metric norm of its velocity is at least a fixed multiple of $\sqrt{m'(R^2)}|\dot R|$, and radial paths attain this bound up to a fixed constant. Since $\mathrm{d} R=\mathrm{d} s/(2\sqrt{s})$, this gives (10.22.5). The metric is smooth and positive on compact subsets away from the origin, so divergence of the integral is equivalent to completeness of the metric on $\mathbb P^ n\setminus \{ 0\} $.

On the blowup, apply proposition 15.23.2 and use the squared Jacobian $|z|^{2(n-1)}$. The closed non-pluripolar product is the extension by zero of the pullback density, has mass $\lambda ^ n=\operatorname {vol}([\theta ])$, and is unique in its full-mass class. By theorem 13.14.3, the extended operator agrees with it on potentials that are continuous off $E$, even when the residual tends to $-\infty $ along $E$. Sublogarithmic growth gives the residual zero Lelong number, so the coefficient along $E$ is exactly $c$, but it does not give a bounded difference from $c\log |s_ E|^2$. The metric off $E$ is isometric to the metric on $\mathbb P^ n\setminus \{ 0\} $, which proves the same distance criterion.

The inner asymptotics considered below can be glued to the prescribed outer density while keeping (10.22.2): take a small inner amplitude or radius, join through a smooth positive annulus, and add a nonnegative smooth annular bump to supply the remaining mass. Thus each of these asymptotic behaviors occurs for a density satisfying the hypotheses of the theorem. The assertions below concern the total local potential $U$; subtracting a smooth background potential can change a leading smooth term. $\square$

Proof. We use the exact expressions (10.22.3). Each asymptotic formula follows by bounding $D$ between $(1\pm \epsilon )$ times its model and integrating the model; no derivative of an unspecified error term is taken. In the first case $m^ n\sim nC s^{\ell +1}L^{-b}/(\ell +1)$, which yields (10.22.8). Both Hessian eigenvalues are then comparable to $s^{\gamma -1}L^{-b/n}$, and the substitution of blowup coordinates gives the assertions about the metric and the distance. The positive power of $s$ makes the potential Hölder continuous, after the logarithmic factor is absorbed by a slightly smaller exponent.

In the second case, $n\int _0^ s t^{-1}C(\log (1/t))^{-b}\, \mathrm{d} t$ is finite precisely for $b>1$, and it is then asymptotic to $nC L^{1-b}/(b-1)$. This gives (10.22.9). Since $\mathrm{d} U=-m\, \mathrm{d} L$, boundedness of $U$ is equivalent to convergence of $\int ^\infty L^{-a}\, \mathrm{d} L$, that is, to $a>1$. The distance is a constant multiple of $\int ^\infty L^{-(a+1)/2}\, \mathrm{d} L$ and has the same strict threshold. Explicit primitives give the stated finite and divergent behavior, including the logarithm at $a=1$. On the blowup the density is comparable to $|z|^{-2}(\log (1/|z|))^{-b}$; for every $p>1$, its $p$th power acquires an exponentially growing factor in logarithmic coordinates, so the density is not in $L^ p$.

In the third case, $m^ n\sim C L^{-n}(\log L)^{-d}$. The criteria for the potential and for the distance become convergence of

respectively. Their thresholds are $d>n$ and $d>2n$, and direct integration gives the stated tails and divergent behavior. A bounded tail of order a positive power of $\log L$ tends to zero more slowly than every positive power of $|z|$, so the potential is not Hölder continuous. The mass integral is finite for every real $d$ because the principal factor is $L^{-(n+1)}$ with $n\geq 2$, and the same exponential argument excludes $L^ p$ for $p>1$. In particular, all the endpoint cases occur within the prescribed family of theorem 10.22.1. $\square$