Pome

Lemma 19.5.32082A647C2B55

Lemma 19.5.32.082A647C2B55 Let $\beta $ be a big and nef class on $X$. Then

66140D1891EC 19.5.33
\begin{equation} \left\{ y\in \mathbb {R}^{n-1}: (0,y)\in \Delta _{Y_{\bullet }}(\beta )\right\} =\Delta _{Y_1\supseteq \cdots \supseteq Y_ n}(\beta |_{Y_1}). \end{equation}

Proof. Step 1. We first reduce to the case where $\beta $ is a Kähler class.

Take a Kähler class $\alpha $ on $X$. It follows from the volume formula in theorem 19.5.4 that

\[ \Delta _{Y_{\bullet }}(\beta )=\bigcap _{\epsilon >0}\Delta _{Y_{\bullet }}(\beta +\epsilon \alpha ). \]

Since $\beta |_{Y_1}$ is nef, hence pseudo-effective, definition 19.5.24 similarly gives

\[ \Delta _{Y_1\supseteq \cdots \supseteq Y_ n}(\beta |_{Y_1})= \bigcap _{\epsilon >0}\Delta _{Y_1\supseteq \cdots \supseteq Y_ n}(\beta |_{Y_1}+\epsilon \alpha |_{Y_1}). \]

So it suffices to prove (19.5.33) with $\beta +\epsilon \alpha $ in place of $\beta $.

Step 2. Assume that $\beta $ is a Kähler class. The $\supseteq $ direction in (19.5.33) follows from the extension theorem theorem 5.18.18. To prove the other direction, recall that by theorem 19.5.4, for $t>0$ small enough, we have

\[ \left\{ y\in \mathbb {R}^{n-1}: (t,y)\in \Delta _{Y_{\bullet }}(\beta ) \right\} =\Delta _{Y_1\supseteq \cdots \supseteq Y_ n}\left((\beta -t\{ Y_1\} )|_{Y_1}\right). \]

As $t\to 0+$, the right-hand side converges to $\Delta _{Y_1\supseteq \cdots \supseteq Y_ n}(\beta |_{Y_1})$ with respect to the Hausdorff metric as a consequence of theorem 19.5.4, while the left-hand side converges to

\[ \left\{ y\in \mathbb {R}^{n-1}: (0,y)\in \Delta _{Y_{\bullet }}(\beta ) \right\} \]

by 19.7.9. We conclude our assertion. $\square$

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