Proposition 10.18.1.7B0AC7895B1F Let $c>0$ be real and let $E$ be an effective real divisor on $X$. Suppose $u\in \mathrm{PSH}(X,\omega )$ satisfies
\[ \mathrm{MA}^{\mathrm{ext}}_\omega (u)\geq c f\, \mathrm{d}V, \qquad \omega +\mathrm{dd}^{\mathrm{c}}u\geq [E]. \]
Let $\beta _ E$ be a smooth closed representative of the Bott–Chern class of $[E]$, and put $\kappa =\kappa _\omega (\beta _ E)$. Then $b>0$, $0\leq \kappa <1$ and
C6FE80302EDE
10.18.2
\begin{equation} c\leq b(1-\kappa )^ n. \end{equation}
Suppose that $c=b$, and let $(T_ j)$ be a smooth maximizing sequence for $D_ p$, that is, a sequence of smooth strictly positive forms $T_ j\in \mathscr G_\omega $ with $D_ p(T_ j)\to \rho _\omega (p)$. Then
\[ \int _ X\beta _ E\wedge T_ j=\int _ X[E]\wedge T_ j\longrightarrow 0. \]
Every weak cluster current $T$ therefore satisfies $\int _ X\beta _ E\wedge T=0$, and the class of $E$ is not big. If the densities $p k_{T_ j}$ of the chosen sequence are uniformly integrable, every such cluster current maximizes $D_ p$. In particular, this holds for every smooth maximizing sequence when $p\in \mathcal D_\omega ^0$.
Proof.
By theorem 15.28.4, $b\geq c>0$. Let $T\in \mathscr G_\omega $ be smooth and strictly positive. By lemma 13.14.12, the measure
\[ R=(\omega +\mathrm{dd}^{\mathrm{c}}u)\wedge T-c^{1/n}p k_ T\, \mathrm{d}V \]
is positive, and its mass is $1-c^{1/n}D_ p(T)$. The absolutely continuous term gives no mass to the support of $E$, whereas $[E]\wedge T$ is concentrated there. Restricting $R$ to the support of $E$ and using $\omega +\mathrm{dd}^{\mathrm{c}}u\geq [E]$ therefore gives
\[ \int _ X[E]\wedge T\leq R(X)=1-c^{1/n}D_ p(T). \]
Let $\ell _ E$ be a global logarithmic potential, so that $[E]=\beta _ E+\mathrm{dd}^{\mathrm{c}}\ell _ E$ by the Poincaré–Lelong formula. Since $\ell _ E$ is locally integrable and $T$ is smooth with $\mathrm{dd}^{\mathrm{c}}T=0$, Stokes' theorem for currents identifies the left side with $\int _ X\beta _ E\wedge T$. This integral is nonnegative for every such $T$, so $\kappa \geq 0$; since $D_ p(T)>0$, we also get $\kappa <1$. Taking the supremum of $D_ p(T)$, which is $\rho _\omega (p)=b^{-1/n}$, proves (10.18.2).
Now let $c=b$, and let $(T_ j)$ be a smooth maximizing sequence. Then the right side of the preceding bound tends to zero. Since pairing with the fixed smooth form $\beta _ E$ is continuous under weak convergence of currents, every cluster current satisfies $\int _ X\beta _ E\wedge T=0$. Moreover, (10.18.2) with $c=b$ forces $\kappa =0$, so the class of $E$ is not big by corollary 11.6.2. If the densities along the sequence are uniformly integrable, proposition 12.10.6 shows that every cluster current maximizes $D_ p$. When $p\in \mathcal D_\omega ^0$, theorem 12.10.1 gives this uniform integrability for every sequence.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.