The Dolbeault complex in infinite dimensions. III: Sheaf cohomology in Banach spaces. (English) Zbl 0983.32010

The author generalizes to Banach spaces the usual finite dimensional vanishing theorem for the cohomology groups of the sheaf of germs of holomorphic functions. More precisely, let \(X\) be a Banach space, \(F\) a Fréchet space, \({\mathcal F}\) the sheaf of germs of \(F\)-valued holomorphic functions on \(X\) and \(\Omega\) a pseudoconvex open subset of \(X\) then, if \(q\geq 1\) one has \(H^q(\Omega, {\mathcal F})=0\) provided that \(X\) has an unconditional Schauder basis.
This result is of course very fundamental for holomorphy in Banach spaces; it is the key for the cohomology methods in this infinite dimensional setting.
For a general Banach space the classical Dolbeault complex is not an acyclic resolution of \({\mathcal O}\) so Lempert uses the Čech cohomology. Here are the main steps of the proof:
The Schauder basis gives, for each integer \(N\), a splitting of the space \(X=\mathbb{C}^N \oplus Y\). Then, for each \(\alpha\in ]0,1[\), \(\Omega\) can be exhausted by pseudoconvex open subsets \(\Omega_N (\alpha)\) fibering into balls over \(\mathbb{C}^N\). Using a precise control of the solution of the Cech coboundary operator on \(\mathbb{C}^N\) Lempert can prove \(H^q({\mathfrak B},{\mathcal F})|\Omega_N (\alpha)=0\) for a suitable covering \({\mathfrak B}\) of \(\Omega\) (made of balls \(B(x,r)\) such that \(B(x,2r)\subset \Omega)\) and for \(\alpha <1/2.\)
As a consequence of the Runge-type approximation theorem proved in [L. Lempert, Ann. Inst. Fourier 50, No. 2, 423-442 (2000; Zbl 0969.46032)] this vanishing result and the exhaustion of \(\Omega \) by the \(\Omega_N(\alpha)\) give \(H^q({\mathfrak B},{\mathcal F})= 0\).
Using this fact for all the pseudoconvex open subsets Lempert then proves that the existence of a non zero \(f\) in \(H^p(\Omega,{\mathcal F})\) yields a contradiction.
[For part I and II of this paper see J. Am. Math. Soc. 11, 485-520 (1998; Zbl 0904.32014) and ibid. 12, 775-793 (1999; Zbl 0926.32048).].


32C35 Analytic sheaves and cohomology groups
46G20 Infinite-dimensional holomorphy
Full Text: DOI