18.45.10 The limit measure and the ambient metric894EB16B432E

Proof. For $\alpha \in mQ\cap \mathbb Z^ n$ and $q=\alpha /m$, the normalized monomial $z^\alpha e^ m$ gives the probability measure $\mathrm dP_{m,q}=A_ m(\alpha )^{-1} \mathrm e^{m(q\cdot r-F(r))}a(r)\, \mathrm dr\, \mathrm d\tau $ on $T$. For every $\eta >0$ there is $c_\eta >0$ such that $F(r)-q\cdot r+F^*(q)\geq c_\eta $ whenever $|r-r(q)|\geq \eta $, uniformly for $q\in Q$: on a large compact set this follows from the uniqueness of the minima, and outside it from (18.45.5), which is uniform on $Q$. Together with (18.45.9), this gives $P_{m,q}\{ |r-r(q)|\geq \eta \} \leq C m^{n/2}\mathrm e^{-mc_\eta }$. Hence, for every continuous function on $X$, its integral against $P_{m,q}$ differs from its average over the torus $\{ r=r(q)\} $ by $o(1)$, uniformly in $q\in Q$. The identity $d!m^{-d}B_ m\nu =d!m^{-d}\sum _\alpha P_{m,\alpha /m}$ and the lattice Riemann sums prove (18.45.12). When $d<n$, the measure $\mu _ Q$ is carried by the moment shell, which has real dimension $n+d<2n$, so it is singular with respect to smooth volume forms.

For the curvature, note that $\theta +\mathrm{dd}^{\mathrm{c}}E_ Q=\mathrm{dd}^{\mathrm{c}}G_ Q$ on $T$. By uniform strong convexity of $F^*$ on $Q$, the slope $q(r)$ maximizing $q\cdot r-F^*(q)$ over $Q$ is unique. Adding the variational inequalities at $r$ and $s$ gives $(r-s)\cdot (q(r)-q(s))\geq c|q(r)-q(s)|^2$. Thus $q$ is Lipschitz, and comparison of the maximal values gives $\nabla G_ Q=q$. Hence $G_ Q$ is $C^{1,1}$, and its Hessian has rank at most $d$ almost everywhere, since $q(r)\in b+W$. In logarithmic coordinates, convex mollifications of $G_ Q$ converge in $C^1$, their Hessians stay bounded and converge in $L^ p_{\mathrm{loc}}$ for every finite $p$, and their minors converge in $L^1_{\mathrm{loc}}$. Induction on the degree in the Bedford–Taylor definition of products of bounded plurisubharmonic functions, together with integration by parts, identifies the powers of $\mathrm{dd}^{\mathrm{c}}G_ Q$ on $T$ with the measures given by these minors. The non-pluripolar product agrees with them on $T$ and puts no mass on the pluripolar set $X\setminus T$. The bound on the rank therefore proves the vanishing assertion.

When $d=n$, a maximizer in the interior of $Q$ satisfies $r=\nabla F^*(q)$, so $G_ Q=F$ on the set where $\nabla F(r)\in \operatorname {int}Q$. Elsewhere the maximizing slope lies in one of the finitely many facets of $Q$. On the set where it lies in a given facet, its component normal to that facet is constant, so the derivative of this component vanishes at almost every point of the set: at a point of differentiability and of density, a nonzero derivative would force a cone of positive density outside the set. Consequently $\det D^2G_ Q=\boldsymbol 1_{\{ \nabla F(r)\in Q\} }\det D^2F$ almost everywhere. In our normalization, $(\mathrm{dd}^{\mathrm{c}}F)^ n=n!\det D^2F(r)\, \mathrm dr\, \mathrm d\tau $. The change of variables $q=\nabla F(r)$ gives the asserted identity for the top product. $\square$