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Strong generic vanishing and a higher-dimensional Castelnuovo-de Franchis inequality. (English) Zbl 1206.14067

By the classical Castelnuovo-de Franchis theorem it is known that if a surface satisfies the inequality \(p_g(X)< 2q(X)-3\), then the surface admits a fibration to a curve of genus \(\geq 2\). Recall that \(p_g(X)= h^0(\omega _X)\) and \(q(X)=h^1(\omega _X)\) so that the above inequality can be restated as \(\chi (\omega _X )< q(X)-2\).
In the article under review, the authors extend the Castelnuovo-de Franchis inequality to varieties of maximal Albanese dimension (i.e. varieties whose Albanese map is generically finite onto its image) and arbitrary dimension. They show that if \(X\) is a compact Kähler manifold of maximal Albanese dimension such that \(\chi (\omega _X )< q(X)-\dim X\), then \(X\) admits a fibration on to a compact analytic variety \(Y\) of maximal Albanese dimension with \(0<\dim Y <\dim X\). (One may moreover assume that any desingularization \(\tilde Y\) of \(Y\) has general type and \(\chi (\omega _{\tilde Y})>0\).)
Along the way they also give a positive answer to a conjecture of Green and Lazarsfeld on the vanishing of higher direct images of the Poincaré bundles which generalizes the main result of [C. D. Hacon, J. Reine Angew. Math. 575, 173–187 (2004; Zbl 1137.14012)] from the projective to the Kähler manifold setting.
The proof is based on the use of generic vanishing theorems and the Evans-Griffith’s syzygy theorem.

MSC:

14J40 \(n\)-folds (\(n>4\))
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14F17 Vanishing theorems in algebraic geometry
32J27 Compact Kähler manifolds: generalizations, classification

Citations:

Zbl 1137.14012

References:

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