Dolbeault isomorphisms for holomorphic vector bundles over holomorphic fiber spaces and applications (Q1309184)

Let \(M\) be a locally trivial holomorphic fiber space over a paracompact complex manifold \(N\) whose fibers are biholomorphic onto a Stein manifold and \(E \to M\) be a holomorphic vector bundle over \(M\). Let \(\Omega^ r_ M(E)\) be the sheaf of germs of holomorphic \(r\)-forms with values in \(E\), \({\mathcal F}\) be the sheaf of germs of \(C^ \infty\) functions in \(M\) which is holomorphic along the fibers, \({\mathcal F}^{r,p}\) be the sheaf of germs of \((r,p)\)-forms with coefficients in \({\mathcal F}\) and \({\mathcal F}^{r,p}(E):={\mathcal F} ^{r,p} \otimes \Omega^ 0_ M(E)\). The authors get a resolution of sheaves \[ 0 \to \Omega^ r_ M(E) \to{\mathcal F} ^{r,0}(E) \to {\mathcal F}^{r,1}(E) \to \cdots \to{\mathcal F}^{r,q}(E)\to 0,\quad \text{where } q= \dim_ C N. \] They obtain vanishing theorems \(H^ k \bigl( M,{\mathcal F}^{r,p}(E) \bigr) =0\) \((k \geq 1)\) and a Dolbeault isomorphism \[ H^ p\bigl( M, \Omega^ r_ M (E) \bigr) \cong \biggl\{ \varphi \in H^ 0 \bigl( M,{\mathcal F}^{r,p}(E) \bigr);\;\overline\partial \varphi=0\biggl\} / \overline\partial H^ 0 \bigl( M,{\mathcal F}^{r,p-1}(E) \bigr) \quad \text{for } q \geq p \geq 1. \] It is known that any complex Lie group has a fibration with base a complex torus and fiber a Stein group. Using this fact, they apply those results to the calculation of \(H^ p(G,\Omega^ r_ G)\), where \(G\) is any complex Lie group.