Lemma 5.37.209AA5F4F533CD

Proof. For $r=1$ there is nothing to prove, so let $r\geq 2$. Put $J_ i=\mathcal I(u_ i)$ and $J=\sum _ iJ_ i$. By lemma 5.37.18, these sheaves are locally free of rank one. At each point, the order of $J$ is the smallest of the orders of the $J_ i$, and a local generator of an ideal $J_ i$ attaining this order also generates $J$. Consequently the sum map $\bigoplus _ iJ_ i\to J$ is locally split, and its kernel $E$ is a holomorphic vector bundle of rank $r-1$.

Fix $f\in \Gamma (\Omega ,J)$. Gluing local holomorphic lifts of $f$ with a smooth partition of unity gives a smooth section $s$ of $\bigoplus _ iJ_ i$ whose image is $f$. Since $f$ is holomorphic, $\beta =\bar\partial s$ is an $E$-valued $(0,1)$-form. It suffices to solve $\bar\partial v=\beta $ for a section $v$ of $E$, for then $s-v$ is a holomorphic lift of $f$.

We solve this equation using a bundle metric of positive curvature. Choose a smooth strictly subharmonic exhaustion function $\rho \geq 1$ of $\Omega $. The Kähler metric

is complete: it dominates $2\mathrm i\partial \rho \wedge \bar\partial \rho $, so $\rho $ has bounded gradient with respect to it, and therefore every curve leaving all compact subsets has infinite length.

Equip $E$ with a smooth Hermitian metric $h$. Choose a smooth increasing convex function $\chi $ on $[1,+\infty )$ so that the metric $h_\chi =h\mathrm e^{-\chi (\rho )}$ has

These two requirements can be imposed successively on the compact sublevel sets of $\rho $: increase $\chi '$ so that $\chi '(\rho )$ times the positive Hessian of $\rho $ dominates the negative part of the curvature, and increase $\chi $ so that the integral over each shell $\{ m\leq \rho <m+1\} $ is at most $2^{-m}$. Increasing the slopes preserves the inequalities already achieved.

Use the global frame $\mathrm{d}z$ of the canonical bundle $K_\Omega $ to identify $E$-valued $(0,q)$-forms with $(E\otimes K_\Omega ^{-1})$-valued $(1,q)$-forms, and give $K_\Omega ^{-1}$ its flat coordinate metric. Its curvature is zero, and $\omega _*$ dominates the Euclidean metric, so the identified form $\beta $ remains $L^2$. The Bochner–Kodaira inequality for compactly supported forms $w$ of type $(1,1)$ gives

Here we used that there are no $(1,2)$-forms. By completeness, cutoffs and smoothing extend this inequality to the domain of the adjoint. Since $\beta $ is $L^2$, the Cauchy–Schwarz inequality bounds the functional $w\mapsto \langle w,\beta \rangle $ by $\| \beta \| \| \bar\partial ^*w\| $, and the Riesz representation theorem gives a distributional solution $v$ of $\bar\partial v=\beta $. Thus $s-v$ is distributionally holomorphic, hence holomorphic in every bundle chart, and it is the required global lift. No global weighted norm condition is imposed on its components. $\square$