Definition 18.1.257731FA7D8BFD

Definition 18.1.25.7731FA7D8BFD Let $\varphi _ i\in \mathrm{PSH}(X,\theta _ i)$ for $i=1,\ldots ,n$. Write $T_ i=\theta _ i+\mathrm{dd}^{\mathrm{c}}\varphi _ i$ for each $i=1,\ldots ,n$. Take a Kähler form $\omega $ on $X$. We define the mixed volume $\operatorname{vol}(T_1,\ldots ,T_ n)$ as

Note that $\operatorname{vol}(T_1+\epsilon \omega ,\ldots ,T_ n+\epsilon \omega )$ is decreasing as $\epsilon \to 0+$ due to 18.1.23 3, so the limit exists.

Furthermore, $\operatorname{vol}(T_1,\ldots ,T_ n)$ does not depend on the choice of $\omega $, as a consequence of 18.1.23 3.