Summary
In Sect. 7 a local Cauchy-Fantappie formula for non-degenerate strictly q-convex domains in n-dimensional complex manifolds is constructed, which yields local solutions of \(\delta u = {f_{0,r}}\;if\;r \geq n - q\) if r≥n−q (Theorem 7.8). In Sect. 9 we prove 1/2-Hölder estimates for these solutions. In Sect. 8, by means of the formula from Sect. 7, we prove a local uniform approximation theorem for continuous \(H_{1/2 \to 0}^{0,r}\left( {\bar D,E} \right): = Z_{0,r}^0\left( {\bar D,E} \right)/E_{0,r}^{1/2 \to 0}\left( {\bar D,E} \right).\).
In Sect. 12 we introduce the concept of a q-convex extension of a complex manifold X, and prove that, with respect to such extensions, the Dolbeault cohomology classes of order r admit uniquely determined continuations if r≥n−q (Theorem 12.14), where n = dim ℂ X, and can be uniformly approximated if r=n−q−1 (Theorem 12.11 and Corollary 12.12). Then, as a consequence, we obtain the classical Andreotti-Grauert finiteness theorem (Theorem 12.16): If E is a holomorphic vector bundle over an n-dimensional q-convex manifold X, then dim H0, r (X, E) < ∞ for all r≥n−q, where, in the completely q-convex case, even H0, r (X, E) = 0 for all r≥n−q. Also in Sect. 12, we prove the following supplement to Theorem 11.2: If D is a non-degenerate completely q-convex domain in an n-dimensional complex manifold X, and E is a holomorphic vector bundle over X, then \(H_{1/2 \to 0}^{0,r}\left( {\bar D,E} \right) = 0\) for all r≥n−q (Theorem 12.7).
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© 1988 Springer Science+Business Media New York
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Henkin, G.M., Leiterer, J. (1988). The Cauchy-Riemann Equation on q-Convex Manifolds. In: Andreotti-Grauert Theory by Integral Formulas. Progress in Mathematics, vol 74. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-6724-4_3
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DOI: https://doi.org/10.1007/978-1-4899-6724-4_3
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-3413-1
Online ISBN: 978-1-4899-6724-4
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