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FUJITA DECOMPOSITION AND HODGE LOCI
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 4 / July 2020
- Published online by Cambridge University Press:
- 12 November 2018, pp. 1389-1408
- Print publication:
- July 2020
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- Article
- Export citation
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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.
