PomeLemma 11.46.2.3CE4F16CE935 The canonical $u_\alpha $ is $\kappa $-plurisubharmonic and smooth off $o$. Its non-pluripolar measure has a positive density $q_\alpha $ of mass one, in $L^1$ and in no $L^ p$, $p>1$. Distinct $\alpha $ give distinct ordinary types, although all their multiplier ideals are trivial.
11.46 Residual regularity in radial singularity typesDF55B983AF23
DF55B983AF23
Subtracting a prescribed singularity separates regularity of the residual from the pole of the solution. On projective space, central radial mass gives an exact inverse for the equation. A relative density error then controls the residual through two derivatives, even when the density is merely integrable at the pole. The inverse also permits smooth crossing of density zeros away from it. The pole is the one used in lemma 15.30.1; here its ordinary type is fixed while the density varies.
11.46.1 The prescribed ordinary type31F950502C14
Let $k\geq 2$, let $\kappa $ be the Fubini–Study form on $\mathbb P^ k$ with $\int \kappa ^ k=1$, and put $o=[1:0:\cdots :0]$. On the chart $Z_0\ne 0$ write $z_ i=Z_ i/Z_0$ and set
\[ r=\sum _{j=1}^ k|z_ j|^2,\qquad P=\frac r{1+r},\qquad T=1-\log P, \qquad u_\alpha =1-T^\alpha \quad (0<\alpha <1). \]
The ratio $P=\sum _{i=1}^ k|Z_ i|^2/\sum _{i=0}^ k|Z_ i|^2$ is global. Give $u_\alpha $ its canonical value $-\infty $ at $o$ and value zero on the hyperplane $Z_0=0$. Primes denote differentiation in $r$, and $b_\alpha =k(1-\alpha )>0$ throughout this section.
Proof. Plurisubharmonicity, smoothness off the pole, unit mass and the $L^ p$ assertions are those of lemma 15.30.1. To fix the profile used in the inverse below, put
81B05A10FA0A
11.46.3
\begin{equation} \Phi _\alpha =\log (1+r)+u_\alpha ,\qquad Q_\alpha =r\Phi _\alpha '=P+\alpha (1-P)T^{\alpha -1}. \end{equation}
The tangential and radial eigenvalues are $Q_\alpha /r$ and $Q_\alpha '$, both positive. Radial determinant integration, with the constant fixed by $\Phi =\log (1+r)$, gives
DE57A4B5C703
11.46.4
\begin{equation} \int _{a<|z|^2<b}(\mathrm{dd}^{\mathrm{c}}\Phi )^ k=Q(b)^ k-Q(a)^ k, \qquad Q=r\Phi '. \end{equation}
Since $Q_\alpha (0)=0$ and $Q_\alpha (\infty )=1$, the density is
Smoothness on the hyperplane and the exclusion of the canonical pole in the product leave no further measure. Near $r=0$, with $L=1-\log r$, direct differentiation gives
44CC5DC8E895
11.46.6
\begin{align} Q_\alpha & \sim \alpha L^{\alpha -1},& Q_\alpha '& \sim \frac{\alpha (1-\alpha )}rL^{\alpha -2},\\ (Q_\alpha ^ k)'& \sim \frac{k\alpha ^ k(1-\alpha )}rL^{-1-k(1-\alpha )}.\nonumber \end{align}
In particular $q_\alpha \sim \alpha ^ k(1-\alpha )r^{-k}L^{-1-b_\alpha }$; the radial volume has order $r^{k-1}\, \mathrm{d}r$, recovering these integrability assertions.
The types for different $\alpha $ are ordinarily distinct, since their difference is unbounded at $o$. Yet every $\mathcal I(tu_\alpha )$ is trivial for finite $t>0$: integrability of the constant function reduces near $o$ to $\int ^\infty \mathrm{e}^{-kL+tL^\alpha }\, \mathrm{d}L<\infty $. Ordinary type information here is strictly finer than this ideal information. $\square$
11.46.7 Density criterion and exact inverse6ECAAB69A33B
Here “radial” means depending only on $r$ in the displayed chart. The smoothness assumption along the hyperplane includes a smooth profile in $t=1/r$ near $t=0$.
Theorem 11.46.8.F273E69013C1 Let $q\ge 0$ be radial, smooth on $\mathbb P^ k\setminus \{ o\} $, including the hyperplane at infinity, and satisfy $\int q\kappa ^ k=1$. Suppose that for some $\delta ,A,r_0>0$,
898B5C1F659D
11.46.9
\begin{equation} \left|\frac{q(r)}{q_\alpha (r)}-1\right|\le Ar^\delta \quad (0<r<r_0). \end{equation}
Then there is a genuine radial $\kappa $-psh potential $U$, normalized to have value zero on the hyperplane, with
\[ \mathrm{MA}_\kappa (U)=q\kappa ^ k, \qquad U\sim u_\alpha . \]
Its residual $R=U-u_\alpha $ off $o$ extends continuously across $o$, with value $R_0$, and for small $r$ satisfies
96AA5EC1C8FC
11.46.10
\begin{equation} \begin{aligned} |R-R_0|& \le Cr^\delta L^{\alpha -2},\\ |R'|& \le Cr^{\delta -1}L^{\alpha -2},\qquad |R''|\le Cr^{\delta -2}L^{\alpha -2}. \end{aligned} \end{equation}
It is locally $C^{0,\min \{ 1,2\delta \} }$. For $\delta \ge \frac12$ it extends $C^1$ with zero gradient at $o$; for $\delta \ge 1$ it extends $C^2$ with zero Hessian at $o$. No derivative hypothesis on the error in (11.46.9) is needed for these conclusions. The potential and its residual are smooth on $\mathbb P^ k\setminus \{ o\} $, including every density zero there. The tail condition ensures positive cumulative central mass; local density smoothness alone is not the criterion.
Proof. Let
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11.46.11
\begin{equation} \begin{aligned} M(r)& =\int _{|z|^2<r}q\kappa ^ k,& Q(r)& =M(r)^{1/k},\\ U'(r)& =\frac{Q(r)-P(r)}r,& U(\infty )& =0. \end{aligned} \end{equation}
The tail condition makes $q$ positive near $o$, so $M(r)>0$ for every $r>0$. One has $M'=q(P^ k)'\ge 0$, $Q'\ge 0$ and $M(\infty )=1$. Thus the positive root defining $Q$ is smooth for every finite $r>0$, even where $q$ vanishes. This proves smoothness across all finite density zeros without a vanishing-order assumption. In $t=1/r$, the tail $1-M$ is the integral from $0$ to $t$ of the smooth density profile times $-\mathrm{d}(1+t)^{-k}/\mathrm{d}t$. Thus $M$ and $Q$ are smooth there, with $Q(0)=1$ in that coordinate. The equation $\mathrm{d}U/\mathrm{d}t=-(Q-P)/t$ is smooth there, since $Q-P$ vanishes at $t=0$. This both defines the normalization and proves smoothness along the full hyperplane.
8992F455B382
11.46.12
\begin{equation} |M-Q_\alpha ^ k| \le C\int _0^ r s^{\delta -1}(1-\log s)^{-1-b_\alpha }\, \mathrm{d}s \le Cr^\delta L^{-1-b_\alpha }. \end{equation}
The last inequality follows by writing $s=r\mathrm{e}^{-y}$ and bounding $\int _0^\infty \mathrm{e}^{-\delta y}(L+y)^{-1-b_\alpha }\, \mathrm{d}y$. Dividing by $Q_\alpha ^ k\asymp L^{-b_\alpha }$ shows that $Q/Q_\alpha $ tends to one; the mean value theorem for the positive $k$th root then gives
Moreover the exact identity
\[ \frac{Q'}{Q_\alpha '}= \frac q{q_\alpha }\left(\frac{Q_\alpha }{Q}\right)^{k-1} \]
gives $|Q'-Q_\alpha '|\le Cr^{\delta -1}L^{\alpha -2}$. Since $R'=(Q-Q_\alpha )/r$, integration to the origin gives a finite $R_0$ and the first two inequalities in (11.46.10). Differentiating this formula gives the last inequality.
The local potential $\Phi =\log (1+r)+U$ has semipositive Hessian away from $o$ because $Q/r>0$ and $Q'\ge 0$. Also $U=u_\alpha +R_0+o(1)$ near $o$, so $\Phi $ tends to $-\infty $. The local profile has nonnegative increasing slope in $\log r$, so its maxima with its value at $r=\varepsilon $ are plurisubharmonic. They decrease to the locally integrable profile; the local decreasing-limit property gives its genuine plurisubharmonic extension with the canonical pole retained. Thus $U$ is globally $\kappa $-psh. Formula (11.46.4) or the local determinant ratio gives its measure $q\kappa ^ k$ away from $o$; the defining non-pluripolar truncations exclude $o$ and are locally identical to this smooth product elsewhere. The measure has exactly total mass one. The bounded global residual proves the ordinary type.
In real coordinates, with $\rho =|z|=\sqrt r$,
\[ |\nabla R|\le C\rho ^{2\delta -1}L^{\alpha -2}, \qquad |\nabla ^2R|\le C\rho ^{2\delta -2}L^{\alpha -2}. \]
For $2\delta <1$, integration along radial radii bounds differences by $C|\rho _1-\rho _2|^{2\delta }$ and hence by $C|z_1-z_2|^{2\delta }$. For $2\delta \ge 1$ the gradient is bounded, giving local Lipschitz regularity. At $\delta =\frac12$ its bound tends to zero and $R-R_0=o(\rho )$, which give the claimed $C^1$ extension. At $\delta =1$, the Hessian bound tends to zero and $\nabla R=o(\rho )$, giving the $C^2$ extension. Larger $\delta $ only improves these bounds. These arguments justify crossing the pole for $R$; they do not assign a finite value to $U(o)$. $\square$
Among radial genuine solutions smooth off $o$ with this density, no central flux and no hyperplane singularity, the potential is unique up to constants: its flux satisfies $Q^ k=M$, and (11.46.11) then fixes its derivative. No nonsymmetric uniqueness is asserted.
Relative envelope-class membership follows from an explicit stretching family. If $a_* =\inf R$, the genuine $\kappa $-psh functions $v_ t=u_\alpha +t a_*$ satisfy, for every $t\ge 1$,
\[ t^{-1}v_ t+(1-t^{-1})u_\alpha =u_\alpha +a_*\le U. \]
At $t=1$ the term with zero coefficient is omitted, as in definition 16.28.61. Hence $U\in \mathcal E_{\mathrm{env}}(\mathbb P^ k,\kappa ,u_\alpha )$ in the stretching definition, canonically at the common pole.
11.46.14 Smooth residuals from flat errors2C18A3499311
Proposition 11.46.15.33B8FF26FB4B Under the hypotheses of Theorem 11.46.8, suppose $g=q/q_\alpha -1$ satisfies
\[ (r\partial _ r)^ jg=O(r^ N) \quad \text{for every }N,j\ge 0. \]
Then $R-R_0$ extends smoothly across $o$, with every derivative zero there.
Proof. Put $x=-\log r$, $a(x)=Q_\alpha (\mathrm{e}^{-x})^ k$ and $E(x)=M(\mathrm{e}^{-x})-a(x)$. The exact central integral is
\[ E(x)=\int _ x^\infty g(\mathrm{e}^{-y})[-a'(y)]\, \mathrm{d}y, \qquad E'(x)=g(\mathrm{e}^{-x})a'(x). \]
Every derivative of $a$ has at most polynomial growth in $1+x$, and $a\asymp (1+x)^{-b_\alpha }$. The integral bounds $E$ by an exponential of arbitrarily large rate. Repeated differentiation of $E'$ expresses each derivative as a finite Leibniz sum of flat derivatives of $g$ and polynomially bounded derivatives of $a$; all derivatives of $E$ are therefore exponentially flat. Every derivative of $1/a$ has at most polynomial growth, by (11.46.3). Thus $E/a$ has the same flatness. Applying smooth composition near one to
\[ Q-Q_\alpha =Q_\alpha \bigl [(1+E/a)^{1/k}-1\bigr ] \]
proves flatness of every $x$ derivative of $Q-Q_\alpha $. Since $\mathrm{d}R/\mathrm{d}x=-(Q-Q_\alpha )$, integration from $x$ to infinity proves the same for $R-R_0$, including its zeroth derivative. An ordinary $j$th derivative in $r$ is $r^{-j}$ times a polynomial in $\partial _ x$; arbitrarily strong exponential decay absorbs that factor. All ordinary derivatives extend continuously as zero at $r=0$, and repeated one-dimensional integration verifies smoothness there. Composition with $r=|z|^2$ gives the conclusion. $\square$
11.46.16 Compact examples and sharp derivative thresholdsFE6A97B82E18
Proposition 11.46.17.F4F876167D51 The $C^1$ threshold $\delta =1/2$ and $C^2$ threshold $\delta =1$ are sharp. When $2\delta <1$, the stated Hölder exponent is also sharp. Nontrivial probability-density perturbations with smoothly flat residuals exist.
Proof. Choose smooth nonnegative radial cutoffs $\chi _0,\chi _1$ with disjoint supports: $\chi _0=1$ near $r=0$ and vanishes outside a finite small ball; $\chi _1$ is nonzero in a finite annulus. Put
405FC6263C95
11.46.18
\begin{equation} B=\frac{\int r^\delta \chi _0q_\alpha \kappa ^ k}{\int \chi _1q_\alpha \kappa ^ k},\qquad q_ c=q_\alpha \bigl [1+c(r^\delta \chi _0-B\chi _1)\bigr ]. \end{equation}
For sufficiently small $c>0$, $q_ c$ is positive, smooth off $o$ and along the hyperplane, and has total mass exactly one. It is comparable to $q_\alpha $ near $o$, hence is in no $L^ p$, $p>1$. Theorem 11.46.8 gives a compact solution in the prescribed ordinary type. Near $o$ its central flux difference is exactly $c\int _0^ r s^\delta (Q_\alpha ^ k)'\, \mathrm{d}s$. The substitution $s=r\mathrm{e}^{-y}$ and dominated convergence give
\[ \int _0^ r s^{\delta -1}(1-\log s)^{-1-b_\alpha }\, \mathrm{d}s \sim \delta ^{-1}r^\delta L^{-1-b_\alpha }. \]
Using (11.46.6) and the $k$th-root expansion yields
\begin{align*} R_ c'(r)& \sim \frac{c\alpha (1-\alpha )}\delta r^{\delta -1}L^{\alpha -2},\\ R_ c(r)-R_ c(0)& \sim \frac{c\alpha (1-\alpha )}{\delta ^2} r^\delta L^{\alpha -2}. \end{align*}
For $\delta <\frac12$, the radial gradient $2\sqrt r\, R_ c'$ diverges, ruling out $C^1$ extension. For $\delta <1$, a tangential real Hessian eigenvalue $2R_ c'$ diverges, ruling out $C^2$ extension. The annular compensation changes no central flux. For $2\delta <1$ the displayed asymptotic also rules out any larger Hölder exponent. The critical derivative thresholds in Theorem 11.46.8 are therefore sharp for this family.
Replacing $r^\delta \chi _0$ in (11.46.18) by $\mathrm{e}^{-1/r}\chi _0$ and adjusting $B$ by the same integral formula gives a nonzero mass-preserving perturbation satisfying every flat-error hypothesis. Proposition 11.46.15 then supplies a smoothly flat residual across the pole. $\square$
11.46.19 Smooth residuals with an open density-zero annulus309EDE002FAE
Proposition 11.46.20.7D9147EC36A1 For every $k\ge 2$ and $0<\alpha <1$, there is a nonnegative radial probability density $q_0$, smooth off $o$ and along the hyperplane, which equals $q_\alpha $ near $o$ and vanishes on a nonempty open annulus. Its solution has ordinary type $u_\alpha $ and a globally smooth residual, constant near $o$.
Proof. Choose a smooth radial $h$ with $0\le h\le 1$, equal to zero on a nonempty finite annulus and equal to one near $o$ and outside a larger finite annulus. Choose a nonzero nonnegative smooth radial cutoff $\chi $ supported in a disjoint finite annulus where $h=1$. Set
\[ C=\frac{\int (1-h)q_\alpha \kappa ^ k}{\int \chi q_\alpha \kappa ^ k}>0, \qquad q_0=q_\alpha (h+C\chi ). \]
This is nonnegative and has integral one. It is smooth at every point off $o$, including the hyperplane, and equals $q_\alpha $ near $o$. It vanishes on the chosen open annulus and is still in no $L^ p$, $p>1$. Theorem 11.46.8 applies, with zero relative density error on a neighborhood of $o$. There the exact central fluxes agree, so $Q=Q_\alpha $ and $R'=0$. Hence $R$ is constant near $o$ and extends smoothly across it. It is smooth everywhere else by the positive-root inverse, including the entire zero annulus. The solution retains its genuine pole; this smoothness assertion concerns the residual. $\square$
11.46.21 Products with a zero-lower-mass factor31DB1FA6A441
Let $(X_0,h_0)$ be any connected compact Hermitian $m$-fold with $v_-(h_0)=0$, and write $V_0=\int _{X_0}h_0^ m>0$. On $X=X_0\times \mathbb P^ k$ put $N=m+k$ and $H=\operatorname {pr}_0^*h_0+\operatorname {pr}_ Y^*\kappa $, where $\operatorname {pr}_0$ and $\operatorname {pr}_ Y$ are the two projections. In the product formulas below, $U$ and $R$ denote their pullbacks. Corollary 11.46.22.7C248BB2EDD2 The pullback of $U$ has the pulled-back ordinary type and relative envelope-class membership. Its residual retains the proved regularity across $X_0\times \{ o\} $, and its finite Monge–Ampère mass is $\binom Nm V_0$. The background $H$ is nonclosed and $v_-(H)=0$.
Proof. The pullback of $U$ obeys
\[ \mathrm{MA}_ H(U)=\binom Nm\operatorname {pr}_0^*(h_0^ m) \wedge \operatorname {pr}_ Y^*(q\kappa ^ k), \qquad \int _ X\mathrm{MA}_ H(U)=\binom Nm V_0. \]
Indeed the bounded truncation $\max \{ U,-j\} $ is a pullback. The bounded corrected-product expansion of proposition 5.28.2, lemma 5.28.4 leaves only the displayed binomial term by dimensions; multiplication by $1_{\{ U>-j\} }$ and increasing $j$ give the exact formula and finite mass. The residual is the pullback of $R$, so its proved regularity extends across $X_0\times \{ o\} $. It is ordinarily equivalent to the pullback of $u_\alpha $ and belongs to the corresponding relative stretching class by the same explicit $v_ t$.
Also $v_-(H)=0$: bounded $h_0$-psh potentials with masses tending to zero have pullbacks of mass $\binom Nm$ times those masses. The background is genuinely nonclosed, since a closed positive $h_0$ would have all bounded masses equal to $V_0>0$ by Stokes. The zero-lower-mass conclusion retains the stated factor hypothesis; the calculation does not determine an individual density threshold. $\square$
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