Explicit \(\bar\partial\)-primitives of Henkin-Leiterer kernels on Stein manifolds (Q1174485)

Let \(X\) be a Stein manifold of dimension \(n\) and let \(Z(h)\subset X\) be the zero set of a holomorphic map \(h:X\to\mathbb{C}^ p\), \(p\leq n-1\). Let \(K=K(\zeta,z)\) be a Henkin-Leiterer type kernel on \(X\). In particular, \(K\) is a \(\overline\partial\)-closed \((n,n-1)\)-form in \(\zeta\) with singularities at \(\zeta=z\). Because of the \((n-2)\)-completeness of \(X\backslash Z(h)\) there exists, for fixed \(z\in Z(h)\), a \((n,n-2)\)-form \(\eta(\zeta,z)\) on \(X\backslash Z(h)\) with \(\overline\partial_ \zeta\eta(\zeta,z)=K(\zeta,z)\). The author constructs explicitly such \(\overline\partial\)-primitives \(\eta\). They are applied to obtain a representation formula for holomorphic functions (similar to the Cauchy- Fantappiè integral formula) and to prove a theorem on the extendability of \(CR\)-functions.