The Cauchy-Riemann Equation on q-Concave Manifolds

Summary

Sections 13–15 are organized analogously as Chapter II. In Sect. 13 a local Cauchy-Fantappie formula for non-degenerate strictly q-concave domains is constructed, which yields local extension of holomorphic functions (local Hartogs extension phenomenon), as well as local solutions of $\bar \partial {\text{u}}\,{\text{ = }}\,{\text{f}}_{{\text{0,r}}}$ for all l≤r≤q−1. In Sect. 14, first we prove 1/2-Hölder estimates for these local solutions, repeating word for word the arguments from Sect. 9. Then, using the same arguments as in Sect. 11, from these estimates we deduce the following version with uniform estimates of the Andreotti-Grauert finiteness theorem: If D is a non-degenerate strictly q-convex domain in a complex manifold X and E is a holomorphic vector bundle over X, then dim $H_{1/2 \to 0}^{0, r} \left( {\bar D, E} \right) < \infty$ for all 0≤r≤q−1, where $H_{1/2 \to 0}^{0, r} \left( {\bar D, E} \right): = Z_{0, r}^0 \left( {\bar D, E} \right)/E_{0, r}^{1/2 \to 0} \left( {\bar D,E} \right).$. In Sect. 15 we introduce the concept of a q-concave extension of a complex manifold, and prove that the Dolbeault cohomology classes of order 0≤r≤q−1 admit uniquely determined continuations along such extensions (for r=0, this is the global Hartogs extension phenomenon for holomorphic functions). Moreover, corresponding results with uniform estimates are obtained. At the end of Sect. 15 we prove the classical Andreotti-Grauert finiteness theorem: If E is a holomorphic vector bundle over a q-concave manifold X, then dim H^{0, r} (X,E) < ∞ for all 0≤r≤q−1.

Sections 16–19 are devoted to the Dolbeault cohomology of order q of q-concave manifolds. In Sect. 16 we prove that the extensions of such cohomology classes along a q-concave extension are uniquely determined (if exist). Sect. 17 is devoted to the special case of (linearly) concave domains X ⊆ ℂⁿ. Here we establish the Martineau isomorphism between H^{n, n−1} (X) and the space of holomorphic functions on the dual domain of X, where the emphasis is on the boundary behavior of this isomorphism. In Sect. 18 we prove the Andreotti-Norguet theorem. The main assertion of this theorem can be formulated as follows: If D is an n-dimensional complex domain which is both strictly q-concave and strictly (n−q−1)-convex, then dim H^{0, q} (D, E) = ∞. Further, a version with uniform estimates of this theorem is obtained. In Sect. 19 we prove the following version with uniform estimates of the Andreotti-Vesentini separation theorem: For any non-degenerate strictly q-concave domain D in a compact complex manifold X and all holomorphic vector bundles E over X, the space $E_{0, q}^{1/2 \to 0} \left( {\bar D, E} \right)$ is closed with respect to the max-norm. Then, as a consequence, we obtain the classical Andreotti-Vesentini separation theorem.