Lemma 10.8.11F36B5947D011

Proof. In a holomorphic frame of $L$ the local weight $p$ of $h$ is plurisubharmonic, and its restriction to $Y$ is plurisubharmonic, because the restriction is defined and $Y$ is connected. If $Y$ is a point, the value of $p$ there is finite and we take any $\kappa >0$. Otherwise, by the strong openness theorem (corollary 6.11.6), every point of $Y$ has a neighborhood in $Y$ on which $f$ is square integrable with the weight $\mathrm e^{-(1+\kappa )p|_ Y}$ for some $\kappa >0$; since $Y$ is compact, finitely many coordinate products $B'_ i\times B''_ i$ of the form of lemma 10.8.6, with tangential balls slightly larger than needed and each contained in a neighborhood on which (10.8.9) holds, cover $Y$, and one $\kappa >0$ works for all of them.

Lemma 10.8.6, applied in a frame with the plurisubharmonic weight $(1+\kappa )p$, gives holomorphic extensions $F_ i$ of $f$ on these products whose critical integrals with the logarithmic gain are finite. Let $U_ i$ be smaller concentric products that still cover $Y$. On $U_ i$, the smooth factors coming from the frames and the comparison $c|w|^2\leq R_ Y\leq C|w|^2$ of (10.8.9) preserve this finiteness, which gives (10.8.12); for the logarithmic factor we use that $-\log R_ Y\geq c'(-\log |w|^2)$ for a constant $c'>0$, since $R_ Y\leq C|w|^2$ and both $R_ Y$ and $|w|^2$ are less than $\mathrm e^{-4}$.

We now glue the local extensions. Choose smooth functions $\eta _ i$ with compact support in $U_ i$ whose sum is one on a neighborhood of $Y$; they are obtained by grouping the members of a partition of unity with compact supports subordinate to the cover of $X$ by the sets $U_ i$ and $X\setminus Y$ (proposition 5.2.6). Put $G=\sum _ i\eta _ iF_ i$, each term extended by zero. Then $G|_ Y=f$. Since all $F_ i$ are holomorphic with the same restriction $f$, we have $\bar\partial G=\sum _ i(\bar\partial \eta _ i)F_ i$, and on $Y$ this equals $f\, \bar\partial \sum _ i\eta _ i=0$. The bounds on the cutoffs and their derivatives and (10.8.12) give (10.8.13).

Put $\gamma =|\bar\partial G|_{\omega ,h_0}^2$. Since $\bar\partial G$ is smooth and vanishes on $Y$, we have $\gamma \leq C|w|^2$ in normal coordinates near each point of $Y$, hence $\gamma \leq CR_ Y$ near $Y$, by (10.8.9) and the compactness of $Y$. Let $A=\{ R_ Y<\varepsilon \} $; all integrals below are taken with $\mathrm dV$. Hölder's inequality with the exponents $1+\kappa $ and $(1+\kappa )/\kappa $ gives

The first factor is bounded by (10.8.13). In normal polar coordinates, using (10.8.9) and $\gamma =O(R_ Y)$, the integrability of $\gamma R_ Y^{-q}(-\log R_ Y)^{2/\kappa }$ near $Y$ reduces to $\int _0^ r(-\log t)^{2/\kappa }\, \mathrm dt<\infty $. Hence the second factor tends to zero as $\varepsilon \downarrow 0$, by the absolute continuity of the integral, which proves (10.8.14). $\square$