TY - JOUR

T1 - On the rank of the flat unitary summand of the Hodge bundle

AU - González-Alonso, Víctor

AU - Stoppino, Lidia

AU - Torelli, Sara

N1 - Funding information: Received by the editors January 16, 2018, and, in revised form, March 27, 2019. 2010 Mathematics Subject Classification. Primary 14D07, 14D06, 32G20; Secondary 14C30. 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. 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 Econom?a y Competitividad?. 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 germ of this collaboration begun during the interesting Workshop ?Birational Geometry of Surfaces? at the University of Roma Tor Vergata in January 2016. We would like to express our gratitude to Pietro Pirola for giving us the starting kick, and for stimulating and fruitful discussions on the topic. We thank Fabrizio Catanese for his kind interest and for giving us extremely useful suggestions. We are also grateful to Xin Lu for pointing out a gap and an incorrect conjecture in a previous version of this work.

PY - 2019/12/15

Y1 - 2019/12/15

N2 - Let f: S → B be a nonisotrivial fibered surface. We prove that the genus g, the rank uf of the unitary summand of the Hodge bundle f∗ωf, and the Clifford index cf satisfy the inequality uf ≤ g − cf. Moreover, we prove that if the general fiber is a plane curve of degree ≥ 5, then the stronger bound uf ≤ g − cf − 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.

AB - Let f: S → B be a nonisotrivial fibered surface. We prove that the genus g, the rank uf of the unitary summand of the Hodge bundle f∗ωf, and the Clifford index cf satisfy the inequality uf ≤ g − cf. Moreover, we prove that if the general fiber is a plane curve of degree ≥ 5, then the stronger bound uf ≤ g − cf − 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.

UR - http://www.scopus.com/inward/record.url?scp=85075129890&partnerID=8YFLogxK

U2 - 10.48550/arXiv.1709.05670

DO - 10.48550/arXiv.1709.05670

M3 - Article

AN - SCOPUS:85075129890

VL - 372

SP - 8663

EP - 8677

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 12

ER -