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FUJITA DECOMPOSITION AND HODGE LOCI

Part of: Families, fibrations Cycles and subschemes Curves Deformations of analytic structures

Published online by Cambridge University Press:  12 November 2018

Paola Frediani ,
Alessandro Ghigi Open the ORCID record for Alessandro Ghigi [Opens in a new window]  and
Gian Pietro Pirola
Paola Frediani
Affiliation:
Università di PaviaItaly ([email protected]; [email protected]; [email protected])
Alessandro Ghigi
Affiliation:
Università di PaviaItaly ([email protected]; [email protected]; [email protected])
Gian Pietro Pirola
Affiliation:
Università di PaviaItaly ([email protected]; [email protected]; [email protected])
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Abstract

This paper contains two results on Hodge loci in $\mathsf{M}_{g}$. The first concerns fibrations over curves with a non-trivial flat part in the Fujita decomposition. If local Torelli theorem holds for the fibers 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 $\mathsf{M}_{g}$. It is proved that the image under the period map of a divisor in $\mathsf{M}_{g}$ is not contained in a proper totally geodesic subvariety of $\mathsf{A}_{g}$. It follows that a Hodge locus in $\mathsf{M}_{g}$ has codimension at least 2.

Keywords

variations of Hodge structureHodge lociFujita decomposition

MSC classification

Primary: 14C30: Transcendental methods, Hodge theory , Hodge conjecture 14D07: Variation of Hodge structures 14H10: Families, moduli (algebraic) 14H15: Families, moduli (analytic) 14H40: Jacobians, Prym varieties 32G20: Period matrices, variation of Hodge structure; degenerations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 4 , July 2020 , pp. 1389 - 1408
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Copyright
© Cambridge University Press 2018

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Footnotes

The authors were partially supported by MIUR PRIN 2015 ‘Moduli spaces and Lie theory’ and by GNSAGA of INdAM. The first author was also partially supported by FIRB 2012 ‘Moduli Spaces and their Applications’.

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