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A vanishing theorem for Dolbeault cohomology of homogeneous vector bundles. (English) Zbl 0911.14003

Let \(G\) be a connected, complex linear algebraic group and \(P\) a parabolic subgroup. Suppose that \(M\) is a completely reducible \(P\)-module and \(\mathcal{L}(M)\) is the sheaf of sections of the associated holomorphic vector bundle \(G\times^{P}M \to G/P\). Then \(M\) satisfies Dolbeault vanishing if \[ H^{p,q}(G/P,\mathcal{L}(M^{*})) = 0 \] whenever \(q > p\) where \(M^{*}\) is the dual of \(M\). Let \(X^{+}_{P}\) be the weights which occur as heighest weight of some simple \(P\)-module. By definition there is no dominant weight between \(\lambda\) and \(\lambda^{+}\) if \(\lambda < \mu \leq \lambda^{+}\) for \(\mu\in X^{+}_{P}\) implies \(\mu =\lambda^{+}\). Furthermore, by \(V_{\lambda ,P}\) we denote a simple \(P\)-module with heighest weight \(\lambda\). The author’s main result is then:
Theorem: Let \(\lambda\in X^{+}_{P}\). Then there is no dominant weight strictly between \(\lambda\) and \(\lambda^{+}\) if and only if Dolbeault vanishing holds for all \(V_{\mu ,P}\) with \(\mu\in X^{+}_{P}\) satisfying \(\mu\leq\lambda\).
Let \(v_{\lambda}\in V_{\lambda}\) be a heighest weight vector and \(Q\) the stabilizer of \(\mathbb{C} v_{\lambda}\subset V_{\lambda}\). Then, the author shows that \[ H^{p,q}(G/P,\mathcal{L}(V_{\lambda ,P})^{*}) = 0 \] whenever \(q > p\) or \(p+q \geq \dim G/P + \dim(Q/P\cap Q) + \dim V_{\lambda ,P}\).

MSC:

14F17 Vanishing theorems in algebraic geometry
14M17 Homogeneous spaces and generalizations
32L20 Vanishing theorems
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