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$$L^{2}$$ estimates for $$\bar{\partial}$$ across a divisor with Poincaré-like singularities. (English) Zbl 1280.32009
Let $$X$$ be a Stein manifold with Kähler metric $$\omega$$, $$L\rightarrow X$$ a holomorphic line bundle with a (possibly singular) Hermitian metric $$e^{-\varphi}$$, and $$D\subset X$$ a closed smooth complex hypersurface, regarded as a divisor on $$X$$, with associated holomorphic line bundle $$L_D$$ equipped with a smooth Hermitian metric $$e^{-\eta}$$. The operator $$\overline{\partial}$$ is densely defined on the Hilbert space of $$L^2$$ sections of the line bundle $$L\oplus L_D$$ and the author studies the equation $$\overline\partial u =\theta$$ for $$\overline\partial$$-closed $$L\oplus L_D$$-valued $$(0,1)$$-forms $$\theta$$ on $$X$$ with $$L^2_{\text{loc}}$$ coefficients. He introduces specific Hermitian metrics on $$L_D$$ and presents estimates for $$u$$ in terms of $$\theta$$ and these metrics which are sufficient for the existence of an $$L^2_{\text{loc}}$$-solution $$u$$. An emphasis lies on the fact that these estimates are valid under curvature conditions on $$\omega$$, $$\varphi$$ and $$\eta$$ that are less restrictive than those for the usual known $$L^2$$ estimates. Here is one of the two main results: Let $$s, \mu\in \mathbb R$$, $$s\in(0,1)$$, $$\mu\geq 1$$, and $$w\in H^0(X,L_D)$$ with zero divisor $$D$$ and $$\sup_X|w|^2e^{-\eta}=1$$. Assume the curvature condition $$\Theta\leq\min\{dd^c\varphi+$$Ric$$(\omega), dd^c\varphi+$$Ric$$(\omega)-\mu^{-1}dd^c\eta\}$$ for a suitable Kähler form $$\Theta$$ on $$X$$. If $A:=\mu^{1-s}s^{-2}(1-s)^{-1}\int_{X-D}|\theta|_\Theta^2e^{-\varphi}|w|^{-2} (\log(e^{\mu+\eta}|w|^{-2}))^{s-1}dV_\omega<\infty,$ then a solution $$u$$ as above exists and $\int_{X-D}|u|^2e^{-\varphi}|w|^{-2}(\log(e^{\mu+\eta})|w|^{-2})^{s-1}dV_\omega\leq A.$ The proofs are given more or less by direct calculations using twisting techniques elaborated by J. D. McNeal and the author [Ann. Inst. Fourier 57, No. 3, 703–718 (2007; Zbl 1208.32011)].
##### MSC:
 32L05 Holomorphic bundles and generalizations 32F32 Analytical consequences of geometric convexity (vanishing theorems, etc.) 32A36 Bergman spaces of functions in several complex variables
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