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On fibre space structures of a projective irreducible symplectic manifold. (English) Zbl 0932.32027

Topology 38, No. 1, 79-83 (1999); addendum ibid. 40, 431-432 (2001).
A simply connected compact Kähler manifold \(X\) is called irreducible symplectic if the vector space \(H^0(X,\Omega^2_X)\) of holomorphic 2-forms is spanned by a nowhere degenerate 2-form. An irreducible symplectic complex surface is a \(K3\)-surface.
As the main result of the article, a \(2n\)-dimensional projective irreducible symplectic manifold \(X\) admits only very special connected fibrations \(X\to B\) onto a normal projective variety \(B\), \(0< \dim B<\dim X\).
The base space \(B\) has to be \(n\)-dimensional and has at most \(\mathbb{Q}\)-factorial log-terminal singularities. Moreover it is a Fano variety with Picard number equal to one.
The degeneration set of the fibration is at least of codimension 2 in \(X\). The general fiber has trivial canonical bundle and admits some finite unramified covering which is an abelian variety.
In the special case \(n=2\) the fibering is non-degenerate and the general fiber is already an abelian surface.
Of main importance for the proof are results of A. Beauville [J. Differ. Geom. 18, 755-782 (1983; Zbl 0537.53056)] and of A. Fujiki [Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 105-165 (1987; Zbl 0654.53065)] about the existence of special non-degenerate quadratic forms on \(H^2(X,\mathbb{Z})\).

MSC:

32J27 Compact Kähler manifolds: generalizations, classification
14J40 \(n\)-folds (\(n>4\))
14J45 Fano varieties
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