Theorem 12.12.2026E84610646

Proof. We first locate the boundary of $\mathscr P$ along the path. Let $s<t$. Since $\alpha $ is nef, for every $\delta >0$ there is a smooth function $v_\delta $ with $\alpha +\mathrm{dd}^{\mathrm{c}}v_\delta \geq -\delta \Omega $. Taking $\delta <t-s$ makes $a_ s+\mathrm{dd}^{\mathrm{c}}v_\delta $ Kähler, and adding $B\geq 0$ gives a positive representative of $\theta _ s$. The same approximations show that $\theta _ t$ is nef. On the other hand, the hypotheses on $C$ and Stokes' theorem on the compact curve give

for every smooth $v$. At $s=t$ the right side vanishes, which excludes a positive representative of $\theta _ t$. For $s>t$, a representative with $\theta _ s+\mathrm{dd}^{\mathrm{c}}v\geq -\delta \Omega $ and $\delta <s-t$ would contradict this identity, so $\theta _ s$ is not nef.

Since every $a_ s$ with $s\leq t$ is closed and nef, the closed nef formula following theorem 12.11.1 gives $\lambda _\mu (a_ s)=V_ s^{1/n}$, including at $s=t$. The difference $\theta _ t-a_ t=B$ is semipositive, so monotonicity gives $\lambda _\mu (\theta _ t)\geq V^{1/n}$, while any fixed Gauduchon test bounds $\lambda _\mu (\theta _ t)$ from above. Since $\theta _ s=\theta _ t+(t-s)\Omega $, theorem 12.11.1 with the Hermitian form $\Omega $ identifies $\lambda _\mu (\theta _ t)$ with the limit of $\lambda _\mu (\theta _ s)$ as $s\uparrow t$.

Write $f(s)=\lambda _\mu (\theta _ s)$ for $s<t$. The path has constant velocity $-\Omega $, so proposition 12.3.2 shows that $f$ is smooth and

In particular, the ratio $H$ is continuous and independent of the choice of minimizing metrics. For $r<t$, the fundamental theorem of calculus gives

Since $H\geq 0$, letting $r\uparrow t$ and using the limit just established yields the identity in (12.12.3).

It remains to bound $f(0)=\lambda _\mu (\theta _0)$ from above by two test metrics. The Kähler form $\kappa _0$ is Gauduchon, and since $\kappa _0$ is cohomologous to $a_0$, closed Stokes gives $\int _ X\theta _0\wedge \kappa _0^{n-1}=V_0+I_ B$; hence $\lambda _\mu (\theta _0)\leq U_\mu $. For the second test $\gamma _0$, the equation $\gamma _0^ n=V_0\mu $ and the cohomology class of $\gamma _0$ give

so its quotient for $\theta _0$ is $V_0^{1/n}+\Delta _\mu (B)$. Combined with the lower bound $\lambda _\mu (\theta _ t)\geq V^{1/n}$, these two tests prove the upper bound in (12.12.3). If $B\leq M\kappa _0$, closed Stokes yields $\int _ XB\wedge \gamma _0^{n-1}\leq M\int _ X\kappa _0\wedge \gamma _0^{n-1}=MV_0$, which gives the remaining cost bounds.

Finally, $V=\int _ X\alpha ^ n>0$, so the closed nef class of $\alpha $ is big by the nef-and-big criterion on compact Kähler manifolds recalled in [SGPT]. Hence it contains a Kähler current $\alpha +\mathrm{dd}^{\mathrm{c}}\rho \geq \varepsilon \Omega $. Adding $B\geq 0$ gives the last assertion. Unlike the first current, the second is not closed when $\mathrm dB\neq 0$, since its exterior derivative is $\mathrm{d}B$. $\square$