Details
| Original language | English |
|---|---|
| Pages (from-to) | 729-755 |
| Number of pages | 27 |
| Journal | Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni |
| Volume | 33 |
| Issue number | 4 |
| Publication status | Published - 8 Feb 2023 |
Abstract
In this paper, we consider the problem of pointwise determining the fibres of the flat unitary subbundle of a polarized variation of Hodge structures (PVHS) of weight one. Starting from the associated Higgs field, and assuming the base has dimension 1, we construct a family of (smooth but possibly non-holomorphic) morphisms of vector bundles with the property that the intersection of their kernels at a general point is the fibre of the flat subbundle. We explore the first one of these morphisms in the case of a geometric PVHS arising from a family of smooth projective curves, showing that it acts as the cup-product with some sort of “second-order Kodaira–Spencer class” which we introduce, and check in the case of a family of smooth plane curves that this additional condition is non-trivial.
Keywords
- Flat bundle, polarized variation of Hodge structures
ASJC Scopus subject areas
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In: Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni, Vol. 33, No. 4, 08.02.2023, p. 729-755.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Algebraic Geometry
T2 - Punctual characterization of the unitary flat bundle of weight one PVHS and application to families of curves
AU - González-Alonso, Víctor
AU - Torelli, Sara
N1 - Publisher Copyright: © 2023 Accademia Nazionale dei Lincei Published by EMS Press
PY - 2023/2/8
Y1 - 2023/2/8
N2 - In this paper, we consider the problem of pointwise determining the fibres of the flat unitary subbundle of a polarized variation of Hodge structures (PVHS) of weight one. Starting from the associated Higgs field, and assuming the base has dimension 1, we construct a family of (smooth but possibly non-holomorphic) morphisms of vector bundles with the property that the intersection of their kernels at a general point is the fibre of the flat subbundle. We explore the first one of these morphisms in the case of a geometric PVHS arising from a family of smooth projective curves, showing that it acts as the cup-product with some sort of “second-order Kodaira–Spencer class” which we introduce, and check in the case of a family of smooth plane curves that this additional condition is non-trivial.
AB - In this paper, we consider the problem of pointwise determining the fibres of the flat unitary subbundle of a polarized variation of Hodge structures (PVHS) of weight one. Starting from the associated Higgs field, and assuming the base has dimension 1, we construct a family of (smooth but possibly non-holomorphic) morphisms of vector bundles with the property that the intersection of their kernels at a general point is the fibre of the flat subbundle. We explore the first one of these morphisms in the case of a geometric PVHS arising from a family of smooth projective curves, showing that it acts as the cup-product with some sort of “second-order Kodaira–Spencer class” which we introduce, and check in the case of a family of smooth plane curves that this additional condition is non-trivial.
KW - Flat bundle
KW - polarized variation of Hodge structures
UR - http://www.scopus.com/inward/record.url?scp=85165767374&partnerID=8YFLogxK
U2 - 10.4171/RLM/987
DO - 10.4171/RLM/987
M3 - Article
AN - SCOPUS:85165767374
VL - 33
SP - 729
EP - 755
JO - Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni
JF - Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni
SN - 1120-6330
IS - 4
ER -