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Fibred Kähler and quasi-projective groups. (English) Zbl 1051.32013

Let \(X\) be a compact Kähler manifold, \(C\) a compact Riemann surface of genus \(g\geq 2\) and \(\psi:\pi_1(X)\rightarrow\prod_g\) a non-trivial homomorphism between their fundamental groups.
The author shows that \(\psi\) is induced by a holomorphic fibration \(X\rightarrow C\) if and only if \(\psi\) is surjective and the image of the induced map \(H^1(\prod_g,\mathbb Q)\rightarrow H^1(X,\mathbb Q)\) contains a \(g\)-dimensional maximal isotropic subspace with respect to the bilinear pairing \(H^1(X,\mathbb Q)\times H^1(X,\mathbb Q)\rightarrow H^2(X,\mathbb Q)\). Moreover, \(\psi\) is induced by a holomorphic fibration without multiple fibers if and only if \(\psi\) is surjective with finitely generated kernel.
The proof is based on the isotropic subspace theorem of the author [Invent. Math. 104, No. 2, 263–289 (1991; Zbl 0743.32025)]. As an essential tool the author shows that the group \(\prod_g\) and the free group \(\mathbb F_g\) of rank \(g\) have the property that every normal non-trivial subgroup of infinite index is not finitely generated. For every Zariski-open set \(Y\subset X\) and every non-trivial homomorphism \(\varphi: \pi_1(Y)\rightarrow\mathbb F_g\) the following is true: \(\varphi\) is induced by a holomorphic fibration \(f: Y\rightarrow \widetilde{C}\) onto a quasi-projective curve \(\widetilde{C}\) with first Betti number \(g\) and a surjection \(\pi_1(\widetilde{C})\rightarrow \mathbb F_g\) if and only if \(\varphi\) is surjective and the image of the induced map \(H^1(\mathbb F_g,\mathbb Q)\rightarrow H^1(Y,\mathbb Q)\) is a \(g\)-dimensional maximal isotropic subspace. Moreover \(f\) can be assumed to be without multiple fibers if and only if the kernel of \(\varphi\) is finitely generated.
This theorem extends former results of D. Arapura [J. Algebr. Geom. 6, No. 3, 563–597 (1997; Zbl 0923.14010)], I. Bauer [Int. J. Math. 8, No. 4, 441–450 (1997; Zbl 0896.14008)] and the author [Am. J. Math. 122, No. 1, 1–44 (2000; Zbl 0983.14013)]. As an application the author presents some topological conditions which are sufficient for a quasi-projective surface to admit a proper holomorphic submersion onto a quasi-projective curve.

MSC:

32J27 Compact Kähler manifolds: generalizations, classification
14F35 Homotopy theory and fundamental groups in algebraic geometry
32Q55 Topological aspects of complex manifolds
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