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On branched coverings of some homogeneous spaces. (English) Zbl 0935.14008
Let \(X\), \(Y\) be smooth connected complex varieties of dimension \(n\), and let \(f\colon X\to Y\) be a finite morphism of degree \(d\), namely a branched covering of \(Y\); the sheaf \(f_*{\mathcal O}_X\) is locally free of rank \(d\) and the trace map \(f_*{\mathcal O}_X\to {\mathcal O}_Y\) is surjective. Denote by \({\mathcal E}^*\) the kernel of the trace map: \({\mathcal E}^*\) is a vector bundle of rank \(d-1\) on \(Y\). R. Lazarsfeld [Math. Ann. 249, 153-162 (1980; Zbl 0434.32013)], proved that if \(Y={\mathbb P}^n\) then the dual bundle \({\mathcal E}\) of \({\mathcal E}^*\) is ample. As a consequence of this result, Lazarsfeld obtains the following Barth-Lefschetz type theorem:
The morphism \(f_*: H^i({\mathbb P}^n, {\mathbb C})\to H^i(X,{\mathbb C})\) is an isomorphism for \(i\leq n-d+1\).
Recently O. Debarre [Manuscr. Math. 89, No. 4, 407-425 (1996; Zbl 0922.14033)] has conjectured that \({\mathcal E}\) is ample when \(Y\) is a homogeneous space with Picard number \(1\). In the paper under consideration the authors prove that \({\mathcal E}\) is ample when \(Y=LG_n\), where \(LG_n\) is the Lagrangian Grassmannian of maximal isotropic subspaces of a symplectic space of dimension \(2n\), and also when \(Y\) is a quadric of dimension \(3\leq n\leq 6\). In analogy with the case \(Y={\mathbb P}^n\), they obtain the following Barth-Lefschetz type statement:
If \(f: X\to LG_n\) is a branched covering of degree \(d\), then \(f_*: H^i(LG_n,{\mathbb C})\to H^i(X,{\mathbb C})\) is an isomorphism for \(i\leq n-d+1\).
In addition, they show that if \(Y\) is a homogeneous space not necessarily with Picard number \(1\), then \({\mathcal E}\) is generated by global sections. This gives evidence for a more general conjecture, stating that in this case \({\mathcal E}\) should be \(k\)-ample for a suitable \(k\).
Reviewer: R.Pardini (Pisa)

14E20 Coverings in algebraic geometry
14M17 Homogeneous spaces and generalizations
14F45 Topological properties in algebraic geometry
14M15 Grassmannians, Schubert varieties, flag manifolds
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