Theorem 18.47.7A0E408FAE124

Proof. The diagonal monomials belong to $V_ m$. The upper bound for their norms in (18.47.6) gives

For $0\leq s\leq 2$, the grid point $j/m$ nearest to $s/2$ loses at most $1/(4m^2)$ compared with $g(s)$; outside this interval the maximizing endpoint lies on the grid. This proves (18.47.8) on the torus.

For a monomial with offset one, $(a,b)=(j+1,j)$, the lower bound in (18.47.6) shows that the exponent of its normalized contribution, apart from $-m\varphi _ X$ and constants, is at most both

The band with $(a,b)=(j,j+1)$ is obtained by interchanging the indices. Put

The lower bound for the norms and the count of monomials give

The function $R$ extends continuously across the coordinate divisors away from the poles. In a chart at $p_+$, put $A=|\zeta _1|^2$ and $B=|\zeta _2|^2$. The possibly large term of $R$ satisfies

and the other term is bounded; the same holds at $p_-$. Thus $R\leq C(1+r^{-2})$ near either pole, where $r=|\zeta |$ is the local radius, and $\log R$ belongs to $L^ p$ for every finite $p$ with respect to smooth volume forms.

For $m\geq 2$, all selected sections vanish at the poles. The two diagonal sections $T_1^ mT_2^ m$ and $Z_1^ mZ_2^ m$ have no other common zeros, so $\mathcal B_ m>0$ off the poles. Away from the poles, $P$ and $U_ m$ are continuous, so both inequalities extend from the torus to $X\setminus \{ p_+,p_-\} $. They give the asserted local uniform convergence and convergence in $L^ p$, and differentiation of the $L^1$ limit in the sense of distributions gives the weak convergence of the currents. At the poles both $U_ m$ and $P$ equal $-\infty $ for $m\geq 2$, which also proves the pointwise statement on $X$.

For a fixed $m\geq 2$, the lowest-order terms of the selected sections at a pole include every binary monomial of degree $m-1$, and the sum of their squared moduli is comparable to $r^{2(m-1)}$ in all directions. Hence

so the difference $U_ m-P$ tends to $+\infty $ near each pole. Nevertheless, (18.47.9) gives, uniformly in $m\geq 2$ and $r<1$,

Combined with uniform convergence on compact subsets away from the poles, these bounds give uniform convergence of $\mathrm e^{U_ m}$.

The contribution of each normalized monomial to $B_ m\nu $ is exactly the product of two truncated Gaussian probability measures in $x_1,x_2$, centered at $(a/m,b/m)$, with the Haar probability measures in the angles. Completing squares in (18.47.5) and using its uniform bounds proves concentration at these centers, uniformly over all three bands; for example, the probability of $|x_1-a/m|\geq \eta $ is at most $C_\eta \sqrt m\, \mathrm e^{-m\eta ^2/2}$. Uniform continuity of a test function then reduces the sum to three lattice Riemann sums. Each band contains asymptotically one third of the $N_ m$ monomials, and in each band the centers $(a/m,b/m)$ approach the diagonal point $(t,t)$, $t\in [0,1]$. This proves both positive limits. The total mass of $2m^{-2}B_ m\nu $ is $2(3m+1)/m^2\to 0$, which proves the convergence to zero in total variation.

Off the poles, $\theta +\mathrm{dd}^{\mathrm{c}}P$ is the pullback by $f$ of the current on $\mathbb P^1$ defined by $g$. The function $g$ is $C^{1,1}$, and the local complex Hessian of $g(s)$ has rank at most one. Smooth convex approximations and the product of locally bounded potentials show that its square vanishes, as in the Hessian argument of section 18.45. The non-pluripolar product assigns no mass to the two poles, so the global square is zero. In particular, the positive-mass hypothesis of theorem 18.35.5 fails in this model. $\square$