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Lemma 20.1.40DFC2368D5436

Lemma 20.1.40.DFC2368D5436

  1. 0B43092BD979

    The product in definition 20.1.39 is symmetric in its $n$ variables.

  2. 3FE06BA1D6F2

    Let $\mathbb {D}_1,\ldots ,\mathbb {D}_ n,\mathbb {D}'_1$ be big and nef b-divisors over $X$. Then

    \[ \left(\mathbb {D}_1+\mathbb {D}_1',\ldots ,\mathbb {D}_ n \right)=\left(\mathbb {D}_1,\ldots ,\mathbb {D}_ n \right)+\left(\mathbb {D}_1',\ldots ,\mathbb {D}_ n \right). \]
  3. 6704C2A24387

    Let $\mathbb {D}_1,\ldots ,\mathbb {D}_ n$ be big and nef b-divisors over $X$ and $\lambda \geq 0$. Then

    \[ \left(\lambda \mathbb {D}_1,\ldots ,\mathbb {D}_ n \right)=\lambda \left(\mathbb {D}_1,\ldots ,\mathbb {D}_ n \right). \]
  4. 41087E4D84EF

    Let $\mathbb {D}_1,\ldots ,\mathbb {D}_ n$ and $\mathbb {D}'$ be big and nef b-divisors over $X$ such that $\mathbb {D}_1\leq \mathbb {D}'$. Then

    \[ \left(\mathbb {D}_1,\ldots ,\mathbb {D}_ n \right)\leq \left(\mathbb {D}',\mathbb {D}_2,\ldots ,\mathbb {D}_ n \right). \]

Proof. 1, 2 and 3 follow immediately from the corresponding results for mixed volumes, as proved in proposition 18.1.27.

4 Take $\mathcal{I}$-good non-divisorial closed positive $(1,1)$-currents $T_1,\ldots ,T_ n$ and $T'$ so that $\mathbb {D}(T_ i)=\mathbb {D}_ i$ for all $i=1,\ldots ,n$ and $\mathbb {D}(T')=\mathbb {D}'$. Furthermore, we may assume that the $T_ i$'s and $T'$ are Kähler currents by the perturbation argument.

Let $(T_{i}^ j)_ j$ be a quasi-equisingular approximation of $T_ i$ for $i=2,\ldots ,n$. It follows from theorem 16.9.30 that

\[ \int _ X T_1\wedge \cdots \wedge T_ n=\lim _{j\to \infty } \int _ X T_1\wedge T_2^ j\wedge \cdots \wedge T_ n^ j. \]

It suffices to show that for all $j\geq 1$,

\[ \int _ X T_1\wedge T_2^ j\wedge \cdots \wedge T_ n^ j\leq \int _ X T'\wedge T_2^ j\wedge \cdots \wedge T_ n^ j. \]

Therefore, we have reduced to the case where $T_2,\ldots ,T_ n$ have analytic singularities. After a resolution, we may assume that they have log singularities along $\mathbb {Q}$-divisors. By proposition 18.1.27 5, we can further reduce to the case where $T_2,\ldots ,T_ n$ have bounded local potentials. Perturbing $T_2,\ldots ,T_ n$ by a Kähler form, we may further assume that $\{ T_2\} ,\ldots , \{ T_ n\} $ are Kähler classes. By proposition 18.1.27 4, we finally reduce to the case where $T_2,\ldots ,T_ n$ are Kähler forms. In this case, the assertion follows from the pseudo-effectivity of $\{ T'\} -\{ T_1\} $. $\square$

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