Families of curves with Higgs field of arbitrarily large kernel

Víctor González-Alonso; Sara Torelli

doi:10.1112/blms.12437

Families of curves with Higgs field of arbitrarily large kernel

Víctor González-Alonso^*, Sara Torelli

^*Corresponding author for this work

Institute of Algebraic Geometry

Research output: Contribution to journal › Article › Research › peer review

Abstract

In this article, we consider the flat bundle 𝒰 and the kernel 𝒦 of the Higgs field naturally associated to any (polarized) variation of Hodge structures of weight 1. We study how strict the inclusion 𝒰⊆𝒦 can be in the geometric case. More precisely, for any smooth projective curve 𝐶 of genus 𝑔⩾2 and any 𝑟=0,…,𝑔−1, we construct non-isotrivial deformations of 𝐶 over a quasi-projective base such that rk𝒦=𝑟 and rk𝒰⩽𝑔+12.

Original language	English
Pages (from-to)	493-506
Number of pages	14
Journal	Bulletin of the London Mathematical Society
Volume	53
Issue number	2
DOIs	https://doi.org/10.1112/blms.12437
Publication status	Published - 3 Apr 2021

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10.1112/blms.12437Licence: CC BY-NC-ND 4.0

https://arxiv.org/abs/1812.05891Licence: Arxiv nonexclusive-distrib 1.0

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title = "Families of curves with Higgs field of arbitrarily large kernel",

abstract = "In this article, we consider the flat bundle 풰 and the kernel 풦 of the Higgs field naturally associated to any (polarized) variation of Hodge structures of weight 1. We study how strict the inclusion 풰⊆풦 can be in the geometric case. More precisely, for any smooth projective curve 퐶 of genus 푔⩾2 and any 푟=0,…,푔−1, we construct non-isotrivial deformations of 퐶 over a quasi-projective base such that rk풦=푟 and rk풰⩽푔+12.",

author = "V{\'i}ctor Gonz{\'a}lez-Alonso and Sara Torelli",

note = "Funding information: During the development of this work, V. Gonz{\'a}lez?Alonso was at the Institut f{\"u}r Algebraische Geometrie (Leibniz Universit{\"a}t Hannover). S. Torelli was in Dipartimento di Matematica {\textquoteleft}F. Cassoratti{\textquoteright} (Universit di Pavia), as well as supported by PRIN 2015 Moduli spaces and Lie Theory, INdAM ? GNSAGA, FAR 2016 (PV) Variet{\`a} algebriche, calcolo algebrico, grafi orientati e topologici and a Riemann Fellowship (Leibniz Universit{\"a}t Hannover). We want to thank Gian Pietro Pirola for carefully reading previous versions of this work, and spotting an error in the main proof. We also want to thank Lidia Stoppino, Xin Lu and Anand Deopurkar for some very fruitful discussions and enlightening ideas. Sara Torelli also thanks the Riemann Center and the Institute of Algebraic Geometry of Leibniz Universit?t Hannover for their warm hospitality and support during her stay as Riemann Fellow which originated this?work. Open access funding enabled and organized by Projekt DEAL.",

year = "2021",

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doi = "10.1112/blms.12437",

language = "English",

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AU - González-Alonso, Víctor

AU - Torelli, Sara

N1 - Funding information: During the development of this work, V. González?Alonso was at the Institut für Algebraische Geometrie (Leibniz Universität Hannover). S. Torelli was in Dipartimento di Matematica ‘F. Cassoratti’ (Universit di Pavia), as well as supported by PRIN 2015 Moduli spaces and Lie Theory, INdAM ? GNSAGA, FAR 2016 (PV) Varietà algebriche, calcolo algebrico, grafi orientati e topologici and a Riemann Fellowship (Leibniz Universität Hannover). We want to thank Gian Pietro Pirola for carefully reading previous versions of this work, and spotting an error in the main proof. We also want to thank Lidia Stoppino, Xin Lu and Anand Deopurkar for some very fruitful discussions and enlightening ideas. Sara Torelli also thanks the Riemann Center and the Institute of Algebraic Geometry of Leibniz Universit?t Hannover for their warm hospitality and support during her stay as Riemann Fellow which originated this?work. Open access funding enabled and organized by Projekt DEAL.

PY - 2021/4/3

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N2 - In this article, we consider the flat bundle 풰 and the kernel 풦 of the Higgs field naturally associated to any (polarized) variation of Hodge structures of weight 1. We study how strict the inclusion 풰⊆풦 can be in the geometric case. More precisely, for any smooth projective curve 퐶 of genus 푔⩾2 and any 푟=0,…,푔−1, we construct non-isotrivial deformations of 퐶 over a quasi-projective base such that rk풦=푟 and rk풰⩽푔+12.

AB - In this article, we consider the flat bundle 풰 and the kernel 풦 of the Higgs field naturally associated to any (polarized) variation of Hodge structures of weight 1. We study how strict the inclusion 풰⊆풦 can be in the geometric case. More precisely, for any smooth projective curve 퐶 of genus 푔⩾2 and any 푟=0,…,푔−1, we construct non-isotrivial deformations of 퐶 over a quasi-projective base such that rk풦=푟 and rk풰⩽푔+12.

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JO - Bulletin of the London Mathematical Society

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Families of curves with Higgs field of arbitrarily large kernel

Abstract

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