Pome

Definition 6.14.56A7E913297D5

The same rational-coefficient analytic-singularity definition. The source versions retain their local presentations, type notation, and explanatory footnotes.

Definition 6.14.5.6A7E913297D5 A quasi-plurisubharmonic function $\varphi $ on $X$ has analytic singularities if every point of $X$ has a connected coordinate neighborhood $V$ on which

60D977A9A056 6.14.6
\begin{equation} \varphi =c\log \sum _{i=1}^ r|g_ i|^2+b, \qquad c\in \mathbb Q_{>0},\quad b\text{ bounded}, \end{equation}

where $r\geq 1$ is an integer and $g_1,\ldots ,g_ r\in \mathcal O(V)$ are not all identically zero. It has neat analytic singularities if such presentations exist with $b$ smooth. Let $c\in \mathbb Q_{>0}$ and let $\mathfrak a\subset \mathcal O_ X$ be a coherent ideal sheaf. We say that $\varphi $ has analytic singularities of type $(c,\mathfrak a)$ if presentations (6.14.6) exist with this coefficient $c$, where on each $V$ the functions $g_1,\ldots ,g_ r$ generate $\mathfrak a$.

Read the alternative statement and proof

Definition 6.14.5.6A7E913297D5 We say $\varphi \in \mathrm{QPSH}(X)$ has analytic singularities if for each $x\in X$, we can find a connected open neighborhood $U$ of $x$ such that $\varphi |_ U$ has the form:

90744C0CCE8E 5.18.1
\begin{equation} c\log (|f_1|^2+\cdots +|f_ N|^2)+R, \end{equation}

where $f_1,\ldots ,f_ N$ are holomorphic functions on $U$, $c\in \mathbb {Q}_{>0}$ and $R$ is a bounded function on $U$.1

When $R$ can be taken to be smooth2, we say $\varphi $ has neat analytic singularities.

Suppose that there is a coherent ideal $\mathcal{I}\subseteq \mathcal{O}_ X$ on $X$ such that we can choose $U$ so that the $f_1,\ldots ,f_ N$ can be chosen as the generators of $\Gamma (U,\mathcal{I})$ and $c$ is independent of the choice of $U$. Then we say $\varphi $ has analytic singularities of type $(c,\mathcal{I})$.

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