Backgrounds reached at the maximal time of a Chern–Ricci flow are not better behaved, as far as the relation between positive volume and bigness is concerned. The radial background of section 12.14 occurs, up to a smooth $\mathrm{dd}^{\mathrm{c}}$ change, as the limiting background of a Chern–Ricci flow at its maximal time. The normalized currents annihilating it can also be described explicitly.
We use the notation of section 12.14: $n=m+1\geq 3$, $t=(m-1)/m$, $a=x/t$, and
\[ X=\mathbb P^ m\times \mathbb P^1,\qquad \theta =\omega +a w,\qquad \mu =\omega ^ m\wedge w, \qquad Z=\{ p\} \times \mathbb P^1, \]
where $\omega $ and $w$ are the normalized Fubini–Study forms pulled back from the two factors, so that $\mu $ is a probability form. Start the flow from the Hermitian metric
C648779A3DE9
12.34.9
\begin{equation} \omega _0=(n+1)\omega +(a+2)w. \end{equation}
Theorem 12.34.10.AFAF4566FE5D With the normalization of the Ricci form in 12.34.3, the maximal smooth existence time of the Chern–Ricci flow starting from $\omega _0$ is $\tau =1/(2\pi )$. The limiting background $\alpha _\tau =\omega _0-\tau \operatorname {Ric}(\omega _0)$ is nef and not big, and
\[ \alpha _\tau =\theta +\mathrm{dd}^{\mathrm{c}}\log (a+2),\qquad \mathcal V_\mu (\alpha _\tau )=n. \]
Its variational volume is positive for every smooth strictly positive probability form on $X$.
Proof.
The initial metric is strictly positive, and
\[ \omega _0^ n=n(n+1)^ m(a+2)\mu . \]
With our normalization, the Ricci forms of the normalized Fubini–Study metrics are $2\pi n\omega $ and $4\pi w$. Consequently
\[ \operatorname {Ric}(\omega _0) =2\pi \bigl (n\omega +2w-\mathrm{dd}^{\mathrm{c}}\log (a+2)\bigr ). \]
Write $s$ for the flow time and put $r=2\pi s$. The affine background is
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12.34.11
\begin{equation} \alpha _ s=(n+1-nr)\omega +(a+2-2r)w +r\mathrm{dd}^{\mathrm{c}}\log (a+2). \end{equation}
For $0\leq r<1$, adding $\mathrm{dd}^{\mathrm{c}}$ of the smooth function $-r\log (a+2)$ gives a strictly positive representative: the horizontal coefficient is at least one and the vertical coefficient is at least $2(1-r)>0$. At $r=1$ we obtain the identity $\alpha _\tau =\theta +\mathrm{dd}^{\mathrm{c}}\log (a+2)$. Since the logarithm is a smooth function on $X$, the bijection $u\mapsto u-\log (a+2)$ between the potentials of the two backgrounds preserves nefness and bigness, and Stokes' theorem for Gauduchon metrics shows that the variational volume is unchanged. Theorem 12.14.6 therefore shows that the endpoint background is nef, not big, and of volume $n$.
For $r>1$, choose a nonzero smooth nonnegative function $f$ on the base supported in $\{ a+2-2r<0\} $, a neighborhood of $p$. Its pullback $S=f\omega ^ m$ is a closed semipositive $(n-1,n-1)$-form. For every quasi-plurisubharmonic function $v$, distributional Stokes gives
\[ \int _ X(\alpha _ s+\mathrm{dd}^{\mathrm{c}}v)\wedge S =\int _{\mathbb P^ m}f(a+2-2r)\omega ^ m<0, \]
so $\alpha _ s$ is not even pseudoeffective for $r>1$. At $r=1$, a smooth positive representative would contradict nonbigness. Thus $r=1$ is the supremum of the admissible times, and theorem 12.34.4 gives the smooth flow on $[0,\tau )$ with exactly this lifetime; the reference representatives used above need not solve the flow equation. Finally, proposition 12.15.1 gives positivity of the volume for every smooth positive probability form.
$\square$
Proposition 12.34.12.83512AFDC859 The current $[Z]/2$ belongs to $\mathscr G_{\omega _0}$ and annihilates $\alpha _\tau $. Every $R\in \mathscr G_{\omega _0}$ with $\int _ X\alpha _\tau \wedge R=0$ has its trace probability $\sigma _ R=\omega _0\wedge R$ supported on $Z$. In the coefficient-matrix convention of section 12.7, its matrix is
\[ P_ R=n\Pi _{\mathrm{vert}}\qquad \sigma _ R\text{-almost everywhere}, \]
where $\Pi _{\mathrm{vert}}$ is the $\omega _0$-orthogonal projection onto the tangent line of the fiber $\{ p\} \times \mathbb P^1$. In particular, $P_ R$ has rank one, and the determinant-weighted measure of $R$ vanishes for every smooth positive reference density.
Proof.
Only smooth forms need to be restricted to $Z$, and their restrictions satisfy $\omega _0|_ Z=2w$ and $\alpha _\tau |_ Z=0$. This proves that $[Z]/2$ is normalized and annihilates $\alpha _\tau $.
Let $R$ be a general annihilator. Since $\alpha _\tau $ and $\theta $ differ by $\mathrm{dd}^{\mathrm{c}}$ of a smooth function, Stokes' theorem gives $\int _ X\theta \wedge R=0$. The measure $\theta \wedge R$ is nonnegative, so it vanishes. On every compact set $K\subset X\setminus Z$, the explicit form $\theta $ satisfies $\theta \geq c_ K\omega _0$ for some $c_ K>0$. Thus $\sigma _ R(K)=0$, and a countable exhaustion by compact sets shows that $\sigma _ R$ is supported on $Z$.
Relative to $\omega _0$, the matrix of $R$ is characterized, for smooth test forms $\beta $ with matrix $B_\beta $, by
\[ \beta \wedge R=\frac1n\operatorname {tr}(B_\beta P_ R)\sigma _ R, \qquad P_ R\geq 0,\qquad \operatorname {tr}P_ R=n. \]
At points of $Z$, the form $\theta $ is positive on the horizontal $m$-plane and vanishes on the vertical line. The vanishing of $\theta \wedge R$ therefore forces all horizontal diagonal entries of $P_ R$ to vanish, and the inequality $|(P_ R)_{ij}|^2\leq (P_ R)_{ii}(P_ R)_{jj}$ for positive matrices removes their rows and columns. The remaining vertical entry is $n$, which proves the formula and the vanishing of the determinant.
$\square$
For each such annihilator $R$, the trace identity at the end of 12.34.3 becomes
\[ \int _ X\omega (s)\wedge R=1-2\pi s, \qquad 0\leq s<1/(2\pi ). \]
This is a statement about averaged traces. The support statement uses the explicit semipositive representative $\theta $ of the endpoint class; the vanishing of the determinant alone would not give it. Neither statement concerns pointwise collapse of the metrics, their local convergence, or uniqueness of the trace probability.
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