PomeLemma 5.18.14.D9F25876A735 Assume that $X$ is compact. Let $\varphi \in \mathrm{QPSH}(X)$ be a potential with analytic singularities. The sheaf $\mathcal{I}_{\infty }(\varphi )$ is coherent.
Lemma 5.18.14D9F25876A735
Proof. By theorem 5.18.8, we may find a modification $\pi \colon Y\rightarrow X$ such that $\pi ^*\varphi $ has log singularities. Observe that
\[ \mathcal{I}_{\infty }(\varphi )=\pi _*\mathcal{I}_{\infty }(\pi ^*\varphi ), \]
so we may replace $X$ and $\varphi $ by $Y$ and $\pi ^*\varphi $ and assume that $\varphi $ has log singularities along an effective $\mathbb {Q}$-divisor $D$. We decompose $D$ into its irreducible components:
\[ D=\sum _{i=1}^ N a_ iD_ i. \]
In this case, observe that
\[ \mathcal{I}_{\infty }(\varphi )=\mathcal{O}_ X\left(-\sum _{i=1}^ N\left(\lceil a_ i \rceil D_ i\right)\right) \]
is clearly coherent. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.