Vector bundles on curves coming from variation of Hodge structures

Catanese, Fabrizio; Dettweiler, Michael

doi:10.1142/s0129167x16400012

Cited by 23 publications

(52 citation statements)

References 41 publications

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“…In the case of infinite monodromy the bound 1.3 is false: this follows from Remark 38 of [CD16] and from the construction of Lu [Lu17]. In the first version of the paper this bound was stated as a conjecture.…”

Section: Motivation and Statement Of The Resultsmentioning

confidence: 99%

On the rank of the flat unitary summand of the Hodge bundle

González-Alonso

Stoppino

Torelli

2019

Trans. Amer. Math. Soc.

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Let f : S → B f\colon S\to B be a nonisotrivial fibered surface. We prove that the genus g g , the rank u f u_f of the unitary summand of the Hodge bundle f ∗ ω f f_*\omega _f , and the Clifford index c f c_f satisfy the inequality u f ≤ g − c f u_f \leq g - c_f . Moreover, we prove that if the general fiber is a plane curve of degree ≥ 5 \geq 5 , then the stronger bound u f ≤ g − c f − 1 u_f \leq g - c_f-1 holds. In particular, this provides a strengthening of bounds proved by M. A. Barja, V. González-Alonso, and J. C. Naranjo and by F. F. Favale, J. C. Naranjo, and G. P. Pirola. The strongholds of our arguments are the deformation techniques developed by the first author and by the third author and G. P. Pirola, which display here naturally their power and depth.

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Section: Motivation and Statement Of The Resultsmentioning

confidence: 99%

On the rank of the flat unitary summand of the Hodge bundle

González-Alonso

Stoppino

Torelli

2019

Trans. Amer. Math. Soc.

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“…In particular, the previous result applies to the examples provided in [6], in [5] and also in [7], concerning the construction of fibrations where the monodromy of U is infinite. Corollary 6.3.…”

Section: Proof Of the Main Theoremsmentioning

confidence: 85%

“…Let f : S → B be a fibration as those constructed in[CD14],[CD] and[CD16] with U of not finite monodromy. Then the canonical normal function is not torsion.…”

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confidence: 99%

Massey Products and Fujita decompositions on fibrations of curves

Pirola

Torelli

2019

Collect. Math.

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Let f : S → B be a fibred surface and f * ω S/B = U ⊕A be the second Fujita decomposition of f. We study a Massey product related with variation of the Hodge structure over flat sections of U. We prove that the vanishing of the Massey product implies that the monodromy of U is finite and described by morphisms over a fixed curve. The main tools are a lifting lemma of flat sections of U to closed holomorphic forms of S and two classical results due (essentially) to de Franchis. As applications we find a new proof of a theorem of Luo and Zuo for hyperelliptic fibrations. We also analyze, as for the surfaces constructed by Catanese and Dettweiler, the case when U has not finite monodromy.Abstract. Let f : S → B be a fibration of curves and let f * ω S/B = U ⊕ A be the second Fujita decomposition of f. In this paper we study a kind of Massey products, which are defined as infinitesimal invariants by the cohomology of a curve, in relation to the monodromy of certain subbundles of U.The main result states that their vanishing on a general fibre of f implies that the monodromy group acts faithfully on a finite set of morphisms and is therefore finite. In the last part we apply our result in terms of the normal function induced by the Ceresa cycle. On the one hand, we prove that the monodromy group of the whole U of hyperelliptic fibrations is finite (giving another proof of a result due to Luo and Zuo). On the other hand, we show that the normal function is non torsion if the monodromy is infinite (this happens e.g. in the examples shown by Catanese and Dettweiler).

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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%

“…In Section 4 we describe the Hodge substructure provided by the theorem in two examples due to Catanese and Dettweiler [9].…”

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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Vector bundles on curves coming from variation of Hodge structures

Cited by 23 publications

References 41 publications

On the rank of the flat unitary summand of the Hodge bundle

On the rank of the flat unitary summand of the Hodge bundle

Massey Products and Fujita decompositions on fibrations of curves

Fujita Decomposition and Hodge Loci

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