18.45.13 A rank-one selection with nonclosed sampling8461E7D0D299

Example 18.45.14.9600E351F18B On $(\mathbb P^1)^2$ take $L=\mathcal O(4,4)$, $w=0$, and $Q=\{ (2+t,2-t):-1/2\leq t\leq 1/2\} $. Then $\Lambda =\mathbb Z(1,-1)$, $\mathrm dq_\Lambda =\mathrm dt$, and

The limit $\mu _ Q$ has mass one: it is the average, over $-1/2\leq t\leq 1/2$, of the Haar probability measures on the real tori $\{ |z_1|^2=(2+t)/(2-t),\ |z_2|^2=(2-t)/(2+t)\} $; indeed, these equations describe the inverse gradient map. The moment shell is a compact three-dimensional part of the hypersurface $|z_1z_2|=1$. Writing $c(p)=p\log (p/4)+(4-p)\log ((4-p)/4)$, we have $F^*(2+t,2-t)=2c(2+t)$, which gives the envelope explicitly.

The quotient of two adjacent selected monomials is $z_1/z_2$. In homogeneous coordinates $z_ j=v_ j/u_ j$, the corresponding map $[v_1u_2:u_1v_2]$ to $\mathbb P^1$ is undefined at $(0,0)$ and $(\infty ,\infty )$. Thus the limit arises from radial concentration rather than from a globally defined submersion as in section 18.36.

Let $\omega _0$ be the sum of the two Fubini–Study forms, and set $f=c|z_1|^2/(1+|z_1|^2)$ with a constant $c\ne 0$. Then $\omega =\mathrm e^ f\omega _0$ is a torus-invariant Hermitian form, and $\mathrm d\omega =\mathrm e^ f\mathrm df\wedge \omega _0\ne 0$ wherever $\mathrm df\ne 0$. Sampling with $\nu =\omega ^2/\int _ X\omega ^2$ gives the same limit of mass one, while the curvature remains $\theta =4\omega _0$.