Definition 20.1.5178D74F44C30

Definition 20.1.5.178D74F44C30 A (Weil) b-divisor $\mathbb {D}$ over $X$ is an assignment $(\mathbb {D}_{\pi })_{\pi \colon Y\rightarrow X}$, where $\pi \colon Y\rightarrow X$ runs over all modifications of $X$ such that

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   $\mathbb {D}_{\pi }\in \mathrm{H}^{1,1}(Y,\mathbb {R})$;

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   The classes are compatible under push-forwards: If $\pi '\colon Z\rightarrow X$ and $\pi \colon Y\rightarrow X$ are both modifications of $X$ and $\pi '$ dominates $\pi $ through $g\colon Z\rightarrow Y$, then $g_*\mathbb {D}_{\pi '}=\mathbb {D}_{\pi }$.

We also write $\mathbb {D}_ Y=\mathbb {D}_{\pi }$ if there is no risk of confusion.

Given two Weil b-divisors $\mathbb {D}$ and $\mathbb {D}'$ over $X$, we say $\mathbb {D}\leq \mathbb {D}'$ if for each modification $\pi $ of $X$, we have $\mathbb {D}_{\pi }\leq \mathbb {D}'_{\pi }$. Recall that by definition, this means the class $\mathbb {D}'_{\pi }-\mathbb {D}_{\pi }$ is pseudo-effective.

A Weil b-divisor with $\mathbb {D}\geq 0$ is called pseudo-effective.