Pome

Lemma 18.1.2320316D4886CF

Lemma 18.1.23.20316D4886CF

  1. 4642B46CE130

    The mixed volume is symmetric in its $n$-variables.

  2. 2AE20B98C203

    For any Kähler form $\omega $ on $X$, we have

    7F052390F186 18.1.24
    \begin{equation} \operatorname{vol}\left(T_1,\ldots ,T_ n\right)=\lim _{\epsilon \to 0+} \operatorname{vol}\left(T_1+\epsilon \omega ,\ldots ,T_ n+\epsilon \omega \right). \end{equation}
  3. D321C1B261ED

    Consider closed positive $(1,1)$-currents $T_1,\ldots ,T_ n,T_1'$ on $X$ with positive volumes, then

    7.10
    \begin{equation} \operatorname{vol}(T_1+T_1',T_2,\ldots ,T_ n)=\operatorname{vol}(T_1,T_2,\ldots ,T_ n)+\operatorname{vol}(T_1',T_2,\ldots ,T_ n). \end{equation}
  4. 7787D4E19E46

    Consider closed positive $(1,1)$-currents $T_1,\ldots ,T_ n$ on $X$ with positive volumes and $\lambda > 0$, then

    \[ \operatorname{vol}(\lambda T_1,T_2,\ldots ,T_ n)=\lambda \operatorname{vol}(T_1,T_2,\ldots ,T_ n). \]
  5. DAD3C89C5ACC

    Suppose that $T_1,\ldots ,T_ n,S_1,\ldots ,S_ n$ are closed positive $(1,1)$-currents on $X$ with positive volumes such that $T_ i\preceq _{\mathcal{I}} S_ i$ and $\{ T_ i\} =\{ S_ i\} $ for each $i=1,\ldots ,n$. Then

    7.11
    \begin{equation} \operatorname{vol}(T_1,\ldots ,T_ n)\leq \operatorname{vol}(S_1,\ldots ,S_ n). \end{equation}

Proof. 1 This follows from the symmetry of the mixed non-pluripolar product in (18.1.22), cf. proposition 13.11.4 2.

2 For each $i$ and each $\epsilon >0$, we have

\[ P_{\theta _ i}[\varphi _ i]_{\mathcal{I}}\sim _ P P_{\theta _ i+\epsilon \omega }\left[P_{\theta _ i}[\varphi _ i]_{\mathcal{I}}\right]_{\mathcal{I}}=P_{\theta _ i+\epsilon \omega }[\varphi _ i]_{\mathcal{I}} \]

as a consequence of example 18.1.11. Hence using proposition 16.8.22 and the multi-linearity of the non-pluripolar product proposition 13.11.4 6,

\[ \begin{aligned} & \lim _{\epsilon \to 0+}\operatorname{vol}(T_1+\epsilon \omega ,\ldots ,T_ n+\epsilon \omega )\\ =& \lim _{\epsilon \to 0+}\int _ X \left(\theta _1+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}P_{\theta _1}[\varphi _1]_{\mathcal{I}}\right)\wedge \cdots \wedge \left(\theta _ n+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}P_{\theta _ n}[\varphi _ n]_{\mathcal{I}}\right)\\ =& \operatorname{vol}(T_1,\ldots ,T_ n). \end{aligned} \]

3 We write $T_ i=\theta _ i+\mathrm{dd}^{\mathrm{c}}\varphi _ i$ as before for each $i$ and $T_1'=\theta _1'+\mathrm{dd}^{\mathrm{c}}\varphi _1'$. The potentials $P_{\theta _1}[\varphi _1]_{\mathcal I}$ and $P_{\theta _1'}[\varphi _1']_{\mathcal I}$ are $\mathcal I$-model with positive masses, hence $\mathcal I$-good. By proposition 18.1.17, their sum is $\mathcal I$-good. Moreover, it is $\mathcal I$-equivalent to $\varphi _1+\varphi _1'$. Then thanks to proposition 18.1.17,

\[ P_{\theta _1}[\varphi _1]_{\mathcal{I}}+P_{\theta _1'}[\varphi _1']_{\mathcal{I}}\sim _ P P_{\theta _1+\theta _1'}\left[P_{\theta _1}[\varphi _1]_{\mathcal{I}}+P_{\theta _1'}[\varphi _1']_{\mathcal{I}} \right]_{\mathcal{I}} = P_{\theta _1+\theta _1'}\left[\varphi _1+\varphi _1'\right]_{\mathcal{I}} \]

Thus by proposition 16.8.22, we have

\[ \begin{aligned} & \operatorname{vol}(T_1+T_1',T_2,\ldots ,T_ n)\\ =& \int _ X \left( \theta _1+\theta _1'+\mathrm{dd}^{\mathrm{c}}P_{\theta _1+\theta _1'}[\varphi _1+\varphi _1']_{\mathcal{I}} \right)\wedge \left( \theta _2+\mathrm{dd}^{\mathrm{c}}P_{\theta _2}[\varphi _2]_{\mathcal{I}}\right)\wedge \cdots \\ & \wedge \left( \theta _ n+\mathrm{dd}^{\mathrm{c}}P_{\theta _ n}[\varphi _ n]_{\mathcal{I}}\right)\\ =& \int _ X \left( \theta _1+\theta _1'+\mathrm{dd}^{\mathrm{c}}P_{\theta _1}[\varphi _1]_{\mathcal{I}}+\mathrm{dd}^{\mathrm{c}}P_{\theta _1'}[\varphi _1']_{\mathcal{I}} \right)\wedge \left( \theta _2+\mathrm{dd}^{\mathrm{c}}P_{\theta _2}[\varphi _2]_{\mathcal{I}}\right)\wedge \cdots \\ & \wedge \left( \theta _ n+\mathrm{dd}^{\mathrm{c}}P_{\theta _ n}[\varphi _ n]_{\mathcal{I}}\right)\\ =& \operatorname{vol}(T_1,T_2,\ldots ,T_ n)+\operatorname{vol}(T_1',T_2,\ldots ,T_ n). \end{aligned} \]

4 This follows from (18.1.22), proposition 16.6.27 1 and proposition 13.11.4 6.

5 Our assertion follows from the monotonicity theorem theorem 13.13.11. $\square$

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