Proposition 20.1.47

Proposition 20.1.47.754B050BFA6C The product in definition 20.1.41 is upper semicontinuous in the following sense. Suppose that $(\mathbb {D}_ i^ j)_{j\in J}$ are nets of nef b-divisors over $X$ with limits $\mathbb {D}_ i$ for each $i=1,\ldots ,n$. Then

7E0BA3EB7A39 20.1.48

\begin{equation} \varlimsup _{j\in J}\left(\mathbb {D}_1^ j,\ldots ,\mathbb {D}_ n^ j \right)\leq \left( \mathbb {D}_1,\ldots ,\mathbb {D}_ n \right). \end{equation}

Proof. Step 1. We first assume that the $\mathbb D_ i^ j$'s and the $\mathbb D_ i$'s are all big.

Choose $\mathcal I$-good non-divisorial currents $T_ i^ j$ and $T_ i$ such that $\mathbb D(T_ i^ j)=\mathbb D_ i^ j$ and $\mathbb D(T_ i)=\mathbb D_ i$. Fix a Kähler form $\omega $ on $X$ and fix $\epsilon >0$. Since $\mathbb D_{i,X}^ j\to \mathbb D_{i,X}$, for all sufficiently large $j$ the class

\[ \mathbb D_{i,X}+\epsilon \{ \omega \} -\mathbb D_{i,X}^ j \]

is Kähler. Choose a Kähler form $\chi _ i^ j$ in this class and set $S_ i^ j=T_ i^ j+\chi _ i^ j$. Then $S_ i^ j$ lies in the fixed class $\mathbb D_{i,X}+\epsilon \{ \omega \} $ and has the same divisorial Lelong numbers as $T_ i^ j$. Hence, by theorem 18.1.42 and proposition 18.1.27 2,

\[ \varlimsup _{j\in J}\operatorname{vol}\Bigl (T_1^ j,\ldots ,T_ n^ j\Bigr )\leq \varlimsup _{j\in J}\operatorname{vol}\Bigl (S_1^ j,\ldots ,S_ n^ j\Bigr )\leq \operatorname{vol}\left(T_1+\epsilon \omega ,\ldots ,T_ n+\epsilon \omega \right). \]

Therefore,

\[ \varlimsup _{j\in J}\Bigl (\mathbb D_1^ j,\ldots ,\mathbb D_ n^ j\Bigr )\leq \Bigl (\mathbb D_1+\epsilon \mathbb D(\omega ),\ldots ,\mathbb D_ n+\epsilon \mathbb D(\omega )\Bigr ). \]

Letting $\epsilon \to 0+$ gives (20.1.48).

Step 2. Next we handle the general case.

Take a Kähler form $\omega $ on $X$. Then by 20.1.44, for any $\epsilon \in (0,1]$, we have

\[ \begin{aligned} \varlimsup _{j\in J}\left(\mathbb {D}_1^ j,\ldots ,\mathbb {D}_ n^ j \right) \leq & \varlimsup _{j\in J}\left(\mathbb {D}_1^ j+\epsilon \mathbb {D}(\omega ),\ldots ,\mathbb {D}_ n^ j+\epsilon \mathbb {D}(\omega ) \right)\\ \leq & \left( \mathbb {D}_1+\epsilon \mathbb {D}(\omega ),\ldots ,\mathbb {D}_ n+\epsilon \mathbb {D}(\omega ) \right)\\ \leq & \left( \mathbb {D}_1,\ldots ,\mathbb {D}_ n \right)+C\epsilon . \end{aligned} \]

Since $\epsilon $ is arbitrary, our assertion follows. $\square$

Proposition 20.1.47`754B050BFA6C`

Comments

Proposition 20.1.47754B050BFA6C

Comments

Proposition 20.1.47`754B050BFA6C`