6.1 Weighted L2 estimates and holomorphic extension89143AD4638F

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Many constructions in the following constructions need a holomorphic function $F$ on a domain $\Omega \subset \mathbb C^ n$ with a prescribed value at a point, or a prescribed restriction to a hyperplane section, together with a bound for the weighted $L^2$ norm $\int _\Omega |F|^2\mathrm e^{-\varphi }\, \mathrm dV$, where $\varphi $ is a plurisubharmonic weight. Gluing by a partition of unity does not preserve holomorphicity: if $f$ is holomorphic near the support of a smooth cutoff function $\chi $, then $\chi f$ is a global smooth function, but $\bar\partial (\chi f)=f\, \bar\partial \chi $ need not vanish. If we can solve $\bar\partial u=f\, \bar\partial \chi $ with an estimate in a weighted $L^2$ norm, then $F=\chi f-u$ is holomorphic. The weight has two tasks: it must make the equation solvable with a bound on $u$, and, where values are prescribed, it must be so singular that the finiteness of the weighted norm forces $u$ to vanish there.

The basic existence theorem is Hörmander's estimate (theorem 6.4.7, theorem 6.4.9). By the Hilbert space duality of section 6.2, a solution of $\bar\partial u=\alpha $ with a norm bound exists as soon as an a priori inequality holds for the adjoint operator. For compactly supported forms this inequality follows from an integration-by-parts identity, the Bochner–Kodaira–Morrey–Kohn identity, in which the complex Hessian of the weight appears as a curvature term. To pass from compactly supported forms to the whole domain of the adjoint without any assumption on the boundary, we work on pseudoconvex open sets and modify the weight near the boundary by means of a smooth strictly plurisubharmonic exhaustion function, whose existence is proved in section 6.3.

In section 6.5 we apply these estimates. Adding the logarithmic pole ${n\log |z-a|^2}$ to the weight prescribes the value of a holomorphic function at the point $a$ (proposition 6.5.1). For extension from a hyperplane, a bounded twist of the weight and an explicit one-variable weight of $\log \log $ type yield the Ohsawa–Takegoshi extension theorem, with a constant that depends only on the width of the domain in the normal direction (theorem 6.5.4). Its consequences for point values and for integrable germs (corollary 6.5.8, corollary 6.5.9) are the analytic input for the multiplier ideals of section 6.6 and the Bergman kernel approximation of section 6.13.

Historical remarks and references for this chapter are collected in subsection 25.1.11.