6.3. Adapted volume forms for klt Kähler pairs [02FG]

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6.3. Adapted volume forms for klt Kähler pairs

We will be briefer since pairs are mainly of interest to MMP practitioners. The key definition for us will be:

Definition 6.7.

A pair $(V,\Delta)$ is klt iff $K_{V}+\Delta$ is $\mathbb{Q}$ -Cartier and if for any log-resolution $\pi:X\to V$ of $(V,\Delta)$ , we have the numerical equivalence of Cartier divisors:

N(K_{X}+\Delta^{\prime})\cong\pi^{*}N(K_{V}+\Delta)+\sum_{E\ exc.}Na_{E}E

with $a_{E}>-1$ , $\Delta^{\prime}$ the proper transform of $\Delta$ in $X$ (same multiplicities) and $N$ is an integer such that $N(K_{V}+\Delta)$ is Cartier.

Thus a variety $V$ has only klt singularities iff $(X,\emptyset)$ is klt.

Let $\beta$ be a local generator of $\mathcal{O}_{V}(N(K_{V}+\Delta))$ . Then $\beta_{V^{reg}}$ can be viewed as a meromorphic N-canonical form with a pole of order $Nd_{i}$ on $E_{i}$ where $\Delta=\sum_{i}d_{i}E_{i}$ is the decomposition of $\Delta$ into prime divisors. Thus $v_{\beta_{V^{reg}}}$ defines a volume form with poles on $V^{reg}$ , namely $v_{\beta_{V^{reg}}}$ is comparable to $\prod_{i}|s_{i}|^{-2d_{i}}d\lambda$ , where $s_{i}$ denotes the canonical section of $\mathcal{O}(E_{i})$ . If $v_{\beta_{V^{reg}}}$ is a finite measure then $d_{i}<1$ , but the converse is not true. We have the following staightforward extension of lemma 6.4:

Lemma 6.8.

Let $j^{\prime}:V-\cup_{i}E_{i}\to V$ be the canonical inclusion. $j^{\prime}_{*}v_{\beta}$ is a well defined Radon measure on $V$ iff $(X,\Delta)$ is klt.

The definition of an adapted measure for a klt pair is left to the reader.