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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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Full Text: Euclid