Pome

Lemma 18.1.311AEDF3A60767

Lemma 18.1.31.1AEDF3A60767 Let $\omega $ be a Kähler form on $X$. Then there is a constant $C>0$ depending only on $X,\omega ,\{ T_1\} ,\ldots ,\{ T_ n\} $ such that

\[ 0\leq \operatorname{vol}(T_1+\epsilon \omega ,\ldots , T_ n+\epsilon \omega )-\operatorname{vol}(T_1,\ldots ,T_ n)\leq C\epsilon \]

for any $\epsilon \in [0,1]$.

Proof. By the multi-linearity established in proposition 18.1.27, we can write

\[ \operatorname{vol}\left(T_1+\epsilon \omega ,\ldots , T_ n+\epsilon \omega \right)-\operatorname{vol}(T_1,\ldots ,T_ n) \]

as a linear combination of the $2^ n-1$-terms:

\[ \epsilon ^ m\operatorname{vol}\left(\omega ^ m, T_{i_1},\ldots ,T_{i_{n-m}} \right) \]

where $J\subseteq \{ 1,\ldots ,n\} $ is a non-empty subset of cardinality $m$, and $\{ i_1,\ldots ,i_{n-m}\} $ is the complement of $J$. It suffices to show that the coefficients are bounded by a constant as above. Take $D\geq 0$ as the minimal non-negative number so that $D\{ \omega \} +\{ T_ i\} $ is nef for each $i=1,\ldots ,n$. Then by the monotonicity theorem theorem 13.13.11,

\[ \operatorname{vol}\left(\omega ^ m, T_{i_1},\ldots ,T_{i_{n-m}} \right)\leq \{ \omega \} ^ m\cap \left( \{ T_{i_1}\} +D\{ \omega \} \right) \cap \cdots \cap \left( \{ T_{i_{n-m}}\} +D\{ \omega \} \right). \]

Our assertion follows. $\square$

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