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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: Víctor González-Alonso, Lidia Stoppino and Sara Torelli
Journal: Trans. Amer. Math. Soc. 372 (2019), 8663-8677
MSC (2010): Primary 14D07, 14D06, 32G20; Secondary 14C30
DOI: https://doi.org/10.1090/tran/7868
Published electronically: July 8, 2019
MathSciNet review: 4029708
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Abstract: Let $ f\colon S\to B$ be a nonisotrivial fibered surface. We prove that the genus $ g$, the rank $ u_f$ of the unitary summand of the Hodge bundle $ f_*\omega _f$, and the Clifford index $ c_f$ satisfy the inequality $ u_f \leq g - c_f$. Moreover, we prove that if the general fiber is a plane curve of degree $ \geq 5$, then the stronger bound $ 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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Additional Information

Víctor González-Alonso
Affiliation: Gottfried Wilhelm Leibniz Universität Hannover, Institut für Algebraische Geometrie, Welfengarten 1, 30167 Hannover, Germany
Email: gonzalez@math.uni-hannover.de

Lidia Stoppino
Affiliation: Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, 27100, Pavia, Italy
Email: lidia.stoppino@unipv.it

Sara Torelli
Affiliation: Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, 27100, Pavia, Italy
Email: sara.torelli02@universitadipavia.it

DOI: https://doi.org/10.1090/tran/7868
Received by editor(s): January 16, 2018
Received by editor(s) in revised form: March 27, 2019
Published electronically: July 8, 2019
Additional Notes: The first author was partially supported by ERC StG 279723 “Arithmetic of algebraic surfaces” (SURFARI) and the project MTM2015-69135-P of the Spanish “Ministerio de Economía y Competitividad”.
The first and second authors wish to thank the Department of Mathematics of Pavia for the invitation and warm hospitality in February 2016.
The second author was partially supported by FAR Uninsubria. The second and third authors are members of G.N.S.A.G.A.–I.N.d.A.M and were partially supported by MIUR (Italy) through PRIN 2012 “Spazi di Moduli e Teoria di Lie ” and PRIN 2015 “Moduli spaces and Lie theory”.
The third author was partially supported by Fondi dottorato Pavia.
Article copyright: © Copyright 2019 American Mathematical Society