The direct image of the relative dualizing sheaf needs not be semiample

Catanese, Fabrizio; Dettweiler, Michael

doi:10.1016/j.crma.2013.12.015

Cited by 24 publications

(38 citation statements)

References 17 publications

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“…The answers to some long standing open questions in algebraic geometry ( [BC08], [CD14],[CD17], [CD16]) turned out to be related with a construction of Hirzebruch of Galois coverings X of the projective plane P 2 (we stick for simplicity to the case of the complex projective plane P 2 C even if the results are more general) with group (Z/n) r , branched over the union L of (r + 1) lines (we call L a configuration of lines since we want the incidence relation of the lines of L to be fixed 1 ).…”

Section: Introductionmentioning

confidence: 99%

Del Pezzo surfaces, rigid line configurations and Hirzebruch–Kummer coverings

Bauer

Catanese

2018

Boll Unione Mat Ital

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Section: Introductionmentioning

confidence: 99%

Del Pezzo surfaces, rigid line configurations and Hirzebruch–Kummer coverings

Bauer

Catanese

2018

Boll Unione Mat Ital

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“…Fujita decomposition says roughly that the Hodge bundle splits as a direct sum of an ample vector bundle and a unitary flat bundle, see [21,28,7,8,9] and Section 3. Let d be the rank of the flat summand in the Fujita decomposition.…”

Section: Introductionmentioning

confidence: 99%

Fujita Decomposition and Hodge Loci

Frediani¹,

Ghigi²,

Pirola³

2018

J. Inst. Math. Jussieu

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This paper contains two results on Hodge loci in Mg. The first concerns fibrations over curves with a non-trivial flat part in the Fujita decomposition. If local Torelli theorem holds for the fibres and the fibration is non-trivial, an appropriate exterior power of the cohomology of the fiber admits a Hodge substructure. In the case of curves it follows that the moduli image of the fiber is contained in a proper Hodge locus. The second result deals with divisors in Mg. It is proved that the image under the period map of a divisor in Mg is not contained in a proper totally geodesic subvariety of Ag. It follows that a Hodge locus in Mg has codimension at least 2.In particular we have the following corollary:Corollary 1.4 (See Corollary 5.14). If g ≥ 3, any Hodge locus of M g has codimension at least 2.The proof of Theorem 1.3 is based on a result of independent interest (Theorem 5.8) that describes the behaviour of a divisor in M g at the boundary. This result is a variation on an argument in [29]. It allows to use induction on g. The case g = 3 follows, as a very special case, from a theorem by Berndt and Olmos [3] on the codimension of totally geodesic submanifolds in symmetric spaces, see 5.10. The inductive step depends on simple Lie theoretic computations showing that H k × H g−k is a maximal totally geodesic submanifold of H g , see Proposition 5.6.Acknowledgements. The authors wish to thank Professor Atsushi Ikeda for introducing them to paper [45]. The second author would like to thank Professor J.S. Milne for interesting information on arithmetic varieties.

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“…In this paper we first begin recalling previous results ( [21], [22], [9], [10]) concerning Fujita's first and second theorem for Kähler fibre spaces over a curve, asserting that the direct image V of the relative dualizing sheaf splits as the direct sum V = A ⊕ Q, where A is ample and Q is unitary flat. Then we focus on our negative answer ( [9], [10]) to a question posed by Fujita 30 years ago: V does not need to be semiample.…”

Section: Introductionmentioning

confidence: 99%

“…Then we focus on our negative answer ( [9], [10]) to a question posed by Fujita 30 years ago: V does not need to be semiample.…”

Section: Introductionmentioning

confidence: 99%

“…In the previous two examples ( [9], [10]) we had n = 7, but we used another method to produce a base change yielding a semistable fibration; as a consequence the degree of the base change that we needed was much larger than 7 (42 in the easier case), and the semistable fibrations were not described with full details. The description we give here was motivated by a question by Fujino, who asked whether we could give a completely explicit example of a semistable fibration satisfying property (4).…”

Section: Introductionmentioning

confidence: 99%

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Vector bundles on curves coming from variation of Hodge structures

Catanese

Dettweiler

2016

Int. J. Math.

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Abstract. Fujita's second theorem for Kähler fibre spaces over a curve asserts that the direct image V of the relative dualizing sheaf splits as the direct sum V = A ⊕ Q, where A is ample and Q is unitary flat. We focus on our negative answer ([9]) to a question by Fujita: is V semiample?We give here an infinite series of counterexamples using hypergeometric integrals and we give a simple argument to show that the monodromy representation is infinite. Our counterexamples are surfaces of general type with positive index, explicitly given as abelian coverings with group (Z/n) 2 of a Del Pezzo surface of degree 5 (branched on a union of lines forming a bianticanonical divisor), and endowed with a semistable fibration with only 3 singular fibres.The simplest such surfaces are the three ball quotients considered in [3], fibred over a curve of genus 2, and with fibres of genus 4.These examples are a larger class than the ones corresponding to Shimura curves in the moduli space of Abelian varieties.

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The direct image of the relative dualizing sheaf needs not be semiample

Cited by 24 publications

References 17 publications

Del Pezzo surfaces, rigid line configurations and Hirzebruch–Kummer coverings

Del Pezzo surfaces, rigid line configurations and Hirzebruch–Kummer coverings

Fujita Decomposition and Hodge Loci

Vector bundles on curves coming from variation of Hodge structures

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