Lemma 5.37.1095217B5924C0

Proof. Step 1: a twisted $\bar\partial $-estimate. Suppose first that $\Omega $ is bounded, that the weights are smooth on a neighborhood of $\overline\Omega $ with $S<1$ there, and that $\Phi =\log S$ is strictly plurisubharmonic. Set $\chi =\phi _1+\phi _2+\psi $ and $A=(\Phi _{j\bar k})$. For a $(0,1)$-form $b$, write $|b|_ A^2$ for its squared norm with respect to the dual of this positive Hermitian matrix. We claim that every $\bar\partial $-closed $(0,1)$-form $b$ for which the right-hand side below is finite admits a solution $u$ of

To prove this, define for $x<0$

The inequality $-x>1-\mathrm e^ x$ gives $\eta >0$. Also $\eta '= -[1-(1-x)\mathrm e^ x]/(1-\mathrm e^ x)^2<0$, whereas $T'>0$, so $\lambda >0$. Direct differentiation gives

Use the weight $h=\mathrm e^{-\chi -T(\Phi )}$ on the trivial line bundle and evaluate the twisting functions at $\Phi $. The twisted curvature is

Let $\Lambda $ denote contraction by the Euclidean Kähler form and $\bar\partial _ h^*$ the weighted adjoint. For a compactly supported $(n,1)$-form $v$, the twisted Bochner identity and the weighted Cauchy–Schwarz inequality give

More explicitly, the cross term is bounded by $\lambda |\bar\partial _ h^*v|^2+\lambda ^{-1}|(\bar\partial \eta )^*v|^2$; the latter term is exactly the commutator contribution of $\mathrm i\lambda ^{-1}\partial \eta \wedge \bar\partial \eta $. The terms of type $(n+1,1)$ vanish, and the remaining connection norm has a favorable sign.

The Hilbert space existence argument applied to (5.37.13) therefore gives

To see the coefficient, project a test form to $\ker \bar\partial $. On this kernel the second term of (5.37.13) vanishes, and the Cauchy–Schwarz inequality in the curvature norm bounds the functional $v\mapsto \langle b,v\rangle _ h$ by $(\int |b|_ A^2h)^{1/2}\| \sqrt{\eta +\lambda }\, \bar\partial _ h^*v\| _ h$. The Riesz representation theorem gives a solution with the asserted norm. The twisting functions are bounded on $\Omega $, since $\Phi $ is smooth and bounded away from both $-\infty $ and $0$. The argument first uses a complete Kähler metric; approximating the Euclidean metric by the Euclidean metric plus small positive multiples of a complete metric gives the Euclidean statement. Under the identification with canonical-bundle-valued forms, the $(n,0)$ norm integral and the curvature-dual $(n,1)$ integral have the same coordinate expressions as above. Since $h=\mathrm e^{-\chi }(1-\mathrm e^\Phi )$, the identities for $T,\eta ,\lambda $ now give (5.37.12).

Step 2: division for smooth weights. Put $v_ i=\mathrm e^{\phi _ i}/S$ and $\ell =(v_1f,v_2f)$. Equip $\mathbb C^2$ with the diagonal metric $(\mathrm e^{-\phi _1},\mathrm e^{-\phi _2})$. Then $\ell $ is orthogonal to the kernel line of the sum map, generated by $(1,-1)$, and $|\ell |^2=|f|^2/S$. Write $\delta =\phi _1-\phi _2$. The Hessian formula gives

Thus $v_1v_2|\bar\partial \delta |_ A^2\leq 1$ and, for $b=\bar\partial (v_1f)$,

Solve (5.37.12) for this $b$ and set $F=\ell -u(1,-1)$. Since $v_1+v_2=1$, the vector $F=(F_1,F_2)$ is holomorphic and its components sum to $f$. The squared norm of $(1,-1)$ is $\mathrm e^{-\phi _1}+\mathrm e^{-\phi _2}=S\mathrm e^{-\phi _1-\phi _2}$. Orthogonality and (5.37.12) consequently give

This proves (5.37.11) for smooth weights.

Step 3: general weights. Exhaust $\Omega $ by bounded pseudoconvex domains with compact closures in $\Omega $. On each of them, convolve the weights on a slightly larger neighborhood and add a sufficiently small positive multiple of $|z|^2$. These smooth plurisubharmonic weights dominate the original weights, converge to them almost everywhere, preserve $S<1$ on the chosen closure, and make $\Phi $ strictly plurisubharmonic. The corresponding integrals $J$ are bounded by the original one. On every compact subset the weights have common upper bounds, so (5.37.11) gives uniform unweighted local $L^2$ bounds for both components. The holomorphic mean-value inequality and a diagonal extraction give holomorphic limits. Almost-everywhere convergence of the weights and Fatou's lemma yield (5.37.11) on $\Omega $, and $F_1+F_2=f$ passes to the limit. $\square$