11.43 Convex-body criteria for varying references7D0D0E9AC5C3

Proof. The Legendre transform of $L$ is $L^*=\ell $ on $\Sigma $ and $+\infty $ outside it. Since $-\log (n+1)\leq \ell \leq 0$,

A finite convex function of the logarithmic moduli is plurisubharmonic: for a smooth convex function the complex Hessian is positive, and convolution gives the general case. On $T\cap \{ Z_ j\ne 0\} $, add the local Fubini–Study potential $\log \sum _ i|Z_ i/Z_ j|^2$; the resulting functions are again convex functions of the logarithmic moduli of the affine coordinates, and they are bounded above by that smooth potential because $h_ K,g_ K\leq L$. Lemma 5.24.7 therefore gives the extensions across the coordinate hyperplanes and the stated bounds. If $p\in \operatorname {int}K$, then $p\in \operatorname {int}\Sigma $, and at $x_ i=\log (p_ i/(1-\sum _ k p_ k))$ the supremum defining $L=\ell ^*$ is attained at $p\in K$, so $g_ K(x)=L(x)$. This proves $\sup \Phi _ K=0$.

For $C\geq \log (n+1)$, let $v_ C=P_{\omega _{\mathrm{FS}}}(\psi _ K+C,0)$, the envelope of the obstacle $\min (\psi _ K+C,0)$. Since $\Phi _ K$ is a competitor, $\Phi _ K\leq v_ C\leq 0$. The obstacle is upper semicontinuous, so $v_ C\leq \psi _ K+C$ everywhere. The compact torus acting by rotations of the coordinates preserves the obstacle, so $v_ C$ is invariant, and on $T$ the function $f=v_ C+L$ is a finite convex function of $x$ with $f\leq L$ and $f\leq h_ K+C$. The second inequality shows that the Legendre transform $f^*$ is $+\infty $ outside $K$, and the first gives $f^*\geq \ell $. Thus

Hence $v_ C\leq \Phi _ K$ on $T$, and since both functions are quasi-plurisubharmonic and $T$ is the complement of an analytic set, $v_ C=\Phi _ K$ on $\mathbb P^ n$. Letting $C\to \infty $ gives $P_{\omega _{\mathrm{FS}}}[\psi _ K]=\Phi _ K$. Since $\psi _ K\leq \Phi _ K\leq \psi _ K+\log (n+1)$, the envelopes of $\min (\Phi _ K+C,0)$ and $\min (\psi _ K+C,0)$ interlace as $C\to \infty $, so also $P_{\omega _{\mathrm{FS}}}[\Phi _ K]=\Phi _ K$.

For the final assertion put $\varepsilon =1-\delta $, $b=\sum _ i p_{*,i}$ and $\chi =p_*\cdot x-\varepsilon L$ on $T$. The inclusion $p_*+\delta \Sigma \subset \Sigma $ implies $p_{*,i}\geq 0$ and $b\leq \varepsilon $, and the extension of $\chi $ to $\mathbb P^ n$ satisfies

Since $h_ K\geq p_*\cdot x+\delta h_\Sigma $ and $L-h_\Sigma \leq \log (n+1)$, one has $\chi -\delta \log (n+1)\leq \Phi _ K$. Thus $\chi -\delta \log (n+1)$ is an $\varepsilon \omega _{\mathrm{FS}}$-plurisubharmonic competitor for $P_{\varepsilon \omega _{\mathrm{FS}}}(\Phi _ K)$, which proves the claim. $\square$

Proof. On $\mathbb P^ n$, the envelope identity is lemma 11.43.2. In the blowup case, every $\pi ^*\omega _{\mathrm{FS}}$-plurisubharmonic function descends to $\mathbb P^ n$. Indeed, off $E_ q$ add a local potential of $\omega _{\mathrm{FS}}$; the resulting plurisubharmonic function is bounded above near $q$, because the original function is bounded above near the compact fiber $E_ q$, and therefore extends across $q$. The pullback of the extension agrees with the original function off $E_ q$, hence everywhere. Pullback is therefore a bijection from $\mathrm{PSH}(\mathbb P^ n,\omega _{\mathrm{FS}})$ onto $\mathrm{PSH}(X,\Theta )$ preserving inequalities and envelopes, and positive scaling preserves them as well.

On projective space, $\Theta =\omega _{\mathrm{FS}}$ is Kähler. For bigness in the blowup case, choose unitary homogeneous coordinates $[W_0:\cdots :W_ n]$, used only in this paragraph, in which $q=[1:0:\cdots :0]$, set

and let $Q:X\to \mathbb P^{n-1}$ be the map induced by the linear projection from $q$. For $0<t<1$,

The smooth part is a Kähler form, since $(\pi ,Q)$ embeds $X$ as the incidence graph in $\mathbb P^ n\times \mathbb P^{n-1}$ and both coefficients are positive. This proves the assertions about the manifold and the form $\Theta $.

For a smooth convex $g$, the coordinates $z_ i=\mathrm{e}^{x_ i/2+\mathrm{i}\vartheta _ i}$ give

For a finite convex $g$, convolution and the continuity of bounded Monge–Ampère products under local uniform convergence identify $(\mathrm{dd}^{\mathrm{c}}g)^ n$ on $T$ in the same way: its image under $z\mapsto x$ is $n!M_ g$, where $M_ g$ is the Alexandrov measure, which assigns to a Borel set of $\mathbb R^ n$ the Euclidean volume of its subgradient image. For $g_ K$, the bound $g_ K\leq h_ K+\log (n+1)$ and a separation argument put every subgradient in $K$. Conversely, if $p\in \operatorname {int}K$ and $p+c\mathbb B\subset K$, where $\mathbb B$ is the closed unit ball of $\mathbb R^ n$ and $c>0$, then $g_ K(x)-p\cdot x\geq c|x|$; this function attains a minimum, so $p$ is a subgradient of $g_ K$ at some point. Consequently

since the boundary of a convex body has zero volume. The analytic complement of $T$ receives no non-pluripolar mass, so this is the full mass. Locality off the exceptional divisor proves the formula after pullback, since the divisor and its image carry no non-pluripolar mass, and homogeneity gives the factor $r^ n$. The reference is locally bounded off the coordinate hyperplanes and their inverse image, an analytic set, hence has small unbounded locus. Finally, scaling by $r$ and pulling back the competitor found at the end of the proof of lemma 11.43.2 gives the last assertion. $\square$

Proof. By the two preceding lemmas, each $\varphi _ j$ and $\varphi $ is an envelope fixed point with small unbounded locus and the stated positive mass, and it satisfies the smaller-background condition. Theorem 11.42.1 therefore gives solvability and the formulas for the scalars and supporting sets.

The common simplex contains a ball $c+s\mathbb B$ of radius $s$, the inradius of $\delta \Sigma $, where $\mathbb B$ is the closed unit ball. The Hausdorff distance gives $P_ j\subset P+d_ j\mathbb B$, while $s\mathbb B\subset P-c$ and convexity give

Exchanging the roles of $P$ and $P_ j$ bounds both volume ratios by $(1+d_ j/s)^ n$. Subtracting the scalar formulas and including the factor $r_ j^ n/r^ n$ gives (11.43.7). The volume bound holds because $\delta \Sigma $ has volume $\delta ^ n/n!$.

Let $\omega _\ell $ be the modulus of uniform continuity of $\ell $ on $\Sigma $. Matching the maximizers in the definitions of $g_{P_ j}$ and $g_ P$ by points at distance at most $d_ j$ gives

This proves local uniform convergence on $T$. The common lower bound $\chi -\delta \log (n+1)\leq \Phi _{P_ j}$ obtained at the end of the proof of lemma 11.43.2 is integrable on $X$, also after pullback, so boundedness of $r_ j$ and dominated convergence off the coordinate hyperplanes give $L^1$ convergence. Finally, the ratio of the normalized exponential densities lies in $[\mathrm{e}^{-2\eta },\mathrm{e}^{2\eta }]$, where $\eta =\| F_ j-F\| _\infty $. This proves (11.43.8) and hence convergence in total variation. $\square$

Proof. The function $p\mapsto \sum _ i\sqrt{p_ i}$ is concave, so $P_ a$ is convex. Set

Then $b+\delta <1$ and $\sum _ i\sqrt{p_{*,i}}=\sqrt{(n+1)/2}>1$. Adding a point of $\delta \Sigma $ preserves the last inequality and keeps the sum of the coordinates below one, so $p_*+\delta \Sigma $ lies in every $P_ a$.

All the standard basis vectors $e_ i$ belong to $P_ a$, and therefore

Hence the reference is locally bounded away from $o$. The Cauchy–Schwarz inequality gives $\sum _ ip_ i\geq a/n$ on $P_ a$. When all $x_ i<0$, this yields $h_{P_ a}(x)\leq (a/n)\max _ i x_ i$, which tends to $-\infty $ as $z\to o$, and lemma 11.43.2 gives the pole at $o$. Local boundedness outside one point implies that all generic Lelong numbers along prime divisors vanish, and pulling back to the blowup at a different point introduces no pole.

To test analyticity, fix positive $\lambda _1,\ldots ,\lambda _ n$ and approach $o$ along $|z_ i|^2=\mathrm{e}^{-R\lambda _ i}$, $R\to \infty $. Writing $y_ i=\sqrt{p_ i}$, the Cauchy–Schwarz inequality gives

where the minimizer satisfies $\sum _ ip_ i\leq a\leq 1$, so that it lies in $P_ a$. Since $L(-R\lambda )\to 0$ and $\Phi _{P_ a}-\psi _{P_ a}$ is bounded, $-\Phi _{P_ a}/R\to d_ a(\lambda )$ along these curves.

Suppose that the type were that of $c\log \sum _ k|f_ k|^2$ near $o$, with $c>0$ and holomorphic germs $f_ k=\sum _\alpha c_{k,\alpha }z^\alpha $. Exponentiating the two inequalities expressing this equivalence and averaging over the arguments of the $z_ i$ gives the positive series

whose exponential decay rate in $R$ must be $d_ a(\lambda )/c$. That rate is the minimum of $\alpha \cdot \lambda $ over the monomials with nonzero coefficients: a minimizing term gives one bound, and convergence at a fixed $R_0$ bounds the series above by a constant times $\mathrm{e}^{-(R-R_0)\min \alpha \cdot \lambda }$. Near any $\lambda $ with positive coordinates, only finitely many monomials can attain the minimum, because these coordinates have a common positive lower bound and any one nonzero coefficient bounds the degree of the minimizing monomials. The rate is therefore locally piecewise linear in $\lambda $. But

This contradiction proves nonanalyticity. The assertion persists on a blowup at a point different from $o$, which does not change a neighborhood of the pole.

The removed corner $C_ a=\{ p\in \mathbb R^ n:p_ i\geq 0,\ \sum _ i\sqrt{p_ i}<\sqrt a\} =aC_1$ lies in $\Sigma $. The substitution $p_ i=y_ i^2$ gives

The last equality follows by iterated integration of monomials over a simplex. Removing the corner from $\Sigma $, whose volume is $1/n!$, and applying (11.43.4) gives (11.43.11). Since $n\geq 2$, one has $\kappa _ n<1$. $\square$

Proof. If $p\in P_{a_ j}\setminus P_1$, then $u=(\sum _ i\sqrt{p_ i})^2\in [a_ j,1)$ and $p/u\in P_1$. Since $|p|\leq \sum _ ip_ i\leq u$, one has $|p/u-p|\leq 1-u\leq 1-a_ j$. Thus $d_ H(P_{a_ j},P_1)\leq 1-a_ j$, and the hypotheses of the theorem hold with the simplex found in the proof of proposition 11.43.10. The mass formula (11.43.11) and the derivative bound $|\mathrm{d}\log (1-\kappa _ n a^ n)/\mathrm{d}a|\leq n\kappa _ n/(1-\kappa _ n)$ give the estimate for the functionals. By (11.43.12), the directional coefficients of the $j$th reference are multiplied by $r_ ja_ j=1-(j+2)^{-2}$, which strictly increases with $j$, so the ordinary types are distinct, while the masses $r_ j^ n(1-\kappa _ n a_ j^ n)$ strictly decrease to $1-\kappa _ n>0$. $\square$

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

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

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. $\square$

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

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. $\square$