Theorem 10.22.1.A86593744D2D Fix $\lambda >0$ and a smooth positive function $D:(0,\infty )\to (0,\infty )$ such that $D(s)=\lambda ^ n/(1+s)^{n+1}$ for all sufficiently large $s$, and
E811914EE11D
10.22.2
\begin{equation} n\int _0^\infty s^{n-1}D(s)\, \mathrm{d} s=\lambda ^ n. \end{equation}
Let $\mu _ D$ be the measure on $\mathbb P^ n$ equal to $D(|y|^2)\omega _{\mathrm E}^ n$ on the affine chart and to zero on the hyperplane at infinity; near that hyperplane it agrees with $(\lambda \omega _{\mathrm{FS}})^ n$. Then the equation $\langle (\lambda \omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q)^ n\rangle =\mu _ D$ has a unique normalized full-mass solution $q$. It is smooth off $y=0$, and on the affine chart its total local potential $\lambda \log (1+|y|^2)+q(y)$ is a radial function $U(s)$ satisfying
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10.22.3
\begin{equation} \begin{aligned} m(s)^ n& =n\int _0^ s t^{n-1}D(t)\, \mathrm{d} t,\qquad m(s)=sU'(s),\\ m'(s)& =\frac{s^{n-1}D(s)}{m(s)^{n-1}}>0. \end{aligned} \end{equation}
The tangential and radial eigenvalues of its Hessian are $m/s$ and $m'$. For any sufficiently small $s_0>0$, the solution is bounded, equivalently has minimal singularities, precisely when
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10.22.4
\begin{equation} \int _0^{s_0}\frac{m(s)}s\, \mathrm{d} s<\infty . \end{equation}
The distance from the origin for the metric $\lambda \omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q$ on $\mathbb P^ n\setminus \{ 0\} $ is finite precisely when
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10.22.5
\begin{equation} \int _0^{s_0}\frac{\sqrt{m'(s)}}{2\sqrt{s}}\, \mathrm{d} s<\infty . \end{equation}
If this integral diverges, the metric on $\mathbb P^ n\setminus \{ 0\} $ is complete.
Let $\pi :X=\operatorname {Bl}_0\mathbb P^ n\to \mathbb P^ n$ be the blowup at the origin, with exceptional divisor $E$, and let $s_ E$ and $\Theta _ E$ be as in section 15.23. For $\theta =\pi ^*(\lambda \omega _{\mathrm{FS}})+c\Theta _ E$ with $c>0$, the full-mass solution on $X$ whose measure is the pullback of $\mu _ D$ has potential $c\log |s_ E|^2+q\circ \pi +C$, generic Lelong number exactly $c$ along $E$, and mass $\lambda ^ n$. In a blowup chart $\pi (z,w)=(z,zw)$, its density relative to the Euclidean volume form of the coordinates $(z,w)$, normalized as above, is
C19D1AD0E901
10.22.6
\begin{equation} |z|^{2(n-1)}D\bigl (|z|^2(1+|w|^2)\bigr ). \end{equation}
Its ordinary singularity type is minimal precisely under (10.22.4). In the unbounded cases, the residual $q\circ \pi $ grows sublogarithmically and has zero generic Lelong number, but the type is not minimal. The metric off $E$ satisfies the same completeness criterion (10.22.5).
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.
$\square$
Pome
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