A second family has a logarithmic pole whose resolution leaves a smooth residual form degenerating along a different exceptional divisor. Let $o=[1:0:\cdots :0]$ as in subsection 11.43.9, and fix $q\ne o$ in $\mathbb P^ n$. On $X=\operatorname {Bl}_ q\mathbb P^ n$, keep $\Theta =\pi ^*\omega _{\mathrm{FS}}$. For $0<t<1$, put
\[ \psi _ t=t\log \frac{\sum _{i=1}^ n|Z_ i|^2}{\sum _{i=0}^ n|Z_ i|^2}, \qquad K_ t=\{ p\in \Sigma :\textstyle \sum _ i p_ i\geq t\} . \]
On $T$, the function $\psi _ t+L$, a potential of $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}\psi _ t$, equals $(1-t)L+t\log \sum _ i\mathrm{e}^{x_ i}$. It differs by a bounded function from
\[ (1-t)h_\Sigma +t\max _ i x_ i=h_{K_ t},\qquad K_ t=(1-t)\Sigma +t\operatorname {conv}(e_1,\ldots ,e_ n). \]
Thus $\psi _ t\sim \psi _{K_ t}\sim \Phi _{K_ t}$, and $n!\operatorname {Vol}_ n(K_ t)=1-t^ n$. We call $\psi _ t$ the prototype of this ordinary type.
Proposition 11.43.15.C9CD848ECB1A Let $\tau :Z\to X$ be the blowup of the point of $X$ above $o$, with exceptional divisor $E_ o$, and denote the inverse image of $E_ q$ in $Z$ again by $E_ q$. There is a smooth semipositive closed form $\alpha _ t$ on $Z$ such that
B2ABC5CB86A3
11.43.16
\begin{equation} (\pi \circ \tau )^*\omega _{\mathrm{FS}} +\mathrm{dd}^{\mathrm{c}}(\psi _ t\circ \pi \circ \tau )=\alpha _ t+t[E_ o]. \end{equation}
It is positive away from $E_ q$, has rank exactly one at every point of $E_ q$, and satisfies $\int _ Z\alpha _ t^ n=1-t^ n>0$. Moreover, for every $u\in \mathrm{PSH}(X,\Theta )$ with the ordinary type of $\pi ^*\Phi _{K_ t}$, in particular for every solution of the equation in this type, the function $u\circ \tau -\psi _ t\circ \pi \circ \tau $ extends from $Z\setminus E_ o$ to a bounded $\alpha _ t$-plurisubharmonic function on $Z$.
Proof.
On $Y=\operatorname {Bl}_ o\mathbb P^ n$, let $\pi _ o$ be the blowdown map and $Q_ o:Y\to \mathbb P^{n-1}$ the map induced by the linear projection from $o$. As in the proof of lemma 11.43.3, the incidence graph embedding shows that
\[ \beta _ t=(1-t)\pi _ o^*\omega _{\mathrm{FS}} +tQ_ o^*\omega _{\mathrm{FS},n-1} \]
is a Kähler form, where $\omega _{\mathrm{FS},n-1}$ is the Fubini–Study form of $\mathbb P^{n-1}$. Since the two centers are distinct, $Z$ is also the blowup $\sigma :Z\to Y$ of the point above $q$. Put $\alpha _ t=\sigma ^*\beta _ t$. The Poincaré–Lelong formula gives (11.43.16); there is no term along $E_ q$, since $\psi _ t$ is locally bounded near $q$.
In a blowup chart, $\sigma $ is $(s,w_2,\ldots ,w_ n)\mapsto (s,sw_2,\ldots ,sw_ n)$. At $s=0$ its differential has rank one and kills the $n-1$ tangent directions of $E_ q$. Positivity of $\beta _ t$ therefore gives the assertion about the rank; off $E_ q$ the map is biholomorphic.
Write $H=\pi _ o^*[\omega _{\mathrm{FS}}]$ and $E=[E_ o]$ for the cohomology classes on $Y$. The current identity gives $[\beta _ t]=H-tE$. Since $H|_{E_ o}=0$, all mixed top intersections vanish. The normal bundle of $E_ o\cong \mathbb P^{n-1}$ is $\mathcal O_{\mathbb P^{n-1}}(-1)$, so $\int _ YE^ n=(-1)^{n-1}$. Therefore
\[ \int _ Z\alpha _ t^ n=\int _ Y(H-tE)^ n=1-t^ n. \]
The first equality holds because $\sigma $ has degree one. Finally, if $u\sim \pi ^*\Phi _{K_ t}\sim \pi ^*\psi _ t$, then $u\circ \tau $ differs from the pulled-back prototype $\psi _ t\circ \pi \circ \tau $ by a bounded function off $E_ o$. By (11.43.16), after adding a local potential of $\alpha _ t$ this bounded difference becomes plurisubharmonic off $E_ o$, and it extends across $E_ o$ by lemma 5.24.7. No Monge–Ampère product of unbounded potentials is involved.
$\square$
Corollary 11.43.17.FB3A15C67170 Let $t_ j,t\in [t_{\min },t_{\max }]\subset (0,1)$ with $t_ j\to t$, and let $r_ j>0$ tend to $r>0$. The references $r_ j\pi ^*\Phi _{K_{t_ j}}$ satisfy the hypotheses of theorem 11.43.6, so in particular the equation is solvable in the limiting ordinary type, and
\[ \sup _ G|B_ j(G)-B(G)| \leq n|\log (r_ j/r)|+ \frac{nt_{\max }^{n-1}}{1-t_{\max }^ n}|t_ j-t|. \]
At continuous tests, all supporting sets consist of the normalized exponential probabilities of that theorem. At the endpoint $t=1$, where the mass vanishes, no equation with a positive bounded density is solvable in the ordinary type of the prototype $\pi ^*\psi _1$.
Proof.
Choose $p_{*,i}=t_{\max }/n$ and $\delta =(1-t_{\max })/2$; then $p_*+\delta \Sigma $ lies in every $K_ t$ with $t\leq t_{\max }$. For $s\geq t$, moving a point of $K_ t$ toward a vertex $e_ i$ until its coordinate sum reaches $s$ gives
\[ d_ H(K_ s,K_ t)\leq \frac{\sqrt2}{1-t_{\max }}|s-t|. \]
Hence the stability theorem applies. Instead of its general bound, the derivative bound for $\log (1-t^ n)$ on $[t_{\min },t_{\max }]$ and the mass formula give the sharper estimate for the functionals.
At $t=1$, the resolved residual form $\alpha _1$ is pulled back from $\mathbb P^{n-1}$, so $\int _ Z\alpha _1^ n=0$. The argument of proposition 11.43.15 shows that every potential of the ordinary type of $\pi ^*\psi _1$ has a bounded $\alpha _1$-plurisubharmonic remainder. Stokes' theorem for bounded products in a closed background gives this remainder mass zero, and positivity forces its Monge–Ampère measure to vanish. Thus no such potential can solve the equation with density $c\mathrm{e}^ F\mathrm{d}V$ for bounded $F$ and $c>0$. In contrast, the endpoint $a=1$ in corollary 11.43.13 has positive mass and remains solvable.
$\square$
Scaling by $r>0$ multiplies both the resolved residual form and the divisor coefficient by $r$. Solvability is obtained on $X$ through the criterion of theorem 11.42.1, without requiring the resolved residual form to be strictly positive. These criteria use closed projective backgrounds and fixed modifications; we do not know analogous convergence criteria for general nonclosed Hermitian backgrounds, for moving modifications, or for unrestricted degenerations of the singularities. Moreover, the supporting probabilities here only identify the limiting functional: we make no claim about convergence of the solutions or of weighted adjoint products.
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