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.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.