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.
$\square$
Pome
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