Ngô support theorem and polarizability of quasi-projective commutative group schemes
type: Item
instance of: scholarly article
title: Ngô support theorem and polarizability of quasi-projective commutative group schemes
zbMATH Open document ID: 1548.14146
author: Giuseppe Ancona
author: Dragos Fratila
published in: Épijournal de Géométrie Algébrique. EPIGA
publication date: 2024-07-23T00:00:00Z
review text: The celebrated decomposition theorem of \textit{A. Beilinson} et al. [Faisceaux pervers. Actes du colloque ``Analyse et Topologie sur les Espaces Singuliers''. Partie I. 2nd edition. Paris: Société Mathématique de France (SMF) (2018; Zbl 1390.14055)] and \textit{M. A. de Cataldo} [IAS/Park City Math. Ser. 24, 1--58 (2017; Zbl 1439.14060)] is an extremely powerful (and widely applied) tool to describe cohomological properties of proper morphisms between algebraic varieties. The support theorem of \textit{B. C. Ngô} [Publ. Math., Inst. Hautes Étud. Sci. 111, 1--169 (2010; Zbl 1200.22011)] provides additional information to the decomposition theorem: for certain proper morphisms called ``weak abelian fibrations'', the support theorem gives a precise description of the supports of the summands that occur in the decomposition theorem. Originally, Ngô's support theorem was conceived to deal with Hitchin fibrations, but its scope is far larger, and many other applications have been found. Notably, the support theorem has been succesfully applied to certain Lagrangian fibrations of hyper-Kähler varieties, and in particular to the computation of the Hodge numbers of hyper-Kähler varieties of OG10-type and OG6-type [\textit{M. A. A. de Cataldo} et al., J. Math. Pures Appl. (9) 156, 125--178 (2021; Zbl 1483.14071); \textit{B. Wu}, Manuscr. Math. 174, No. 3--4, 1015--1042 (2024; Zbl 07873607)], and to establishing the standard conjectures for certain hyper-Kähler tenfolds [\textit{G. Ancona} et al., ``Relative and absolute Lefschetz standard conjectures for some Lagrangian fibrations'', Preprint, \url{arXiv:2304.00978}].\N\NThe precise definition of ``weak abelian fibration'' is quite long and delicate. Traditionally, it was considered that the most delicate part of the definition was the condition of polarizability of the Tate module; typically, in applications of the support theorem this condition was established by somewhat ``ad hoc'' arguments. The merit of the present pape…
reviewed by: Robert Laterveer
Mathematics Subject Classification ID: 14K05
Mathematics Subject Classification ID: 14K20
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 7885171
zbMATH Keywords: group schemes
zbMATH Keywords: abelian varieties
zbMATH Keywords: Hodge theory
zbMATH Keywords: hyper-Kähler varieties
zbMATH Keywords: algebraic cycles
MaRDI profile type: Publication
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DOI: 10.46298/EPIGA.2024.12345
description: scientific article; zbMATH DE number 7885171