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Corollary 20.1.311D8C494FD558

Corollary 20.1.31.1D8C494FD558 The set of nef b-divisors over $X$ with root $\alpha $ can be naturally identified with

\[ \varprojlim _{\omega } \left( \mathcal{G}(\alpha +\omega )/\sim _{\mathcal{I}} \right), \]

where $\omega $ runs over the directed set of Kähler forms on $X$ (with respect to the preorder of reverse domination), and given two Kähler forms $\omega \leq \omega '$ the transition map

\[ \mathcal{G}(\alpha +\omega )/\sim _{\mathcal{I}}\rightarrow \mathcal{G}(\alpha +\omega ')/\sim _{\mathcal{I}} \]

is induced by the map $\mathcal{G}(\alpha +\omega )\rightarrow \mathcal{G}(\alpha +\omega ')$ sending $T$ to $T+\omega '-\omega $.

Proof. For each Kähler form $\omega $, theorem 20.1.28 identifies $\mathcal G(\alpha +\omega )/\sim _{\mathcal I}$ with the set of big and nef b-divisors whose root on $X$ is $\alpha +\{ \omega \} $. The transition map is exactly addition of $\mathbb D(\omega '-\omega )$. Passing to the inverse limit gives precisely nef b-divisors with root $\alpha $. $\square$

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