Ngô support theorem and polarizability of quasi-projective commutative group schemes (Q6576876)

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 paper is to change this way of thinking, and to show that the polarizability condition is easier to obtain than traditionally thought, thus opening up a large realm of potential applications of Ngô's support theorem. The main result reads as follows:\N\NTheorem: Any quasi-projective commutative group scheme is polarizable. More precisely, if \(L\) is a relatively ample line bundle the first Chern class of \(L\) induces a polarization of the relative Tate module.\N\NThe argument proving this theorem is very natural, and quite surprisingly easy. First, using formal properties of adjunction, one reduces to the absolute case. Next, using a certain ``preservation of ampleness'' (Proposition 6.4 in the paper), one reduces to the case of abelian varieties. For the case of abelian varieties, the pairing obtained is shown to be related to more classical polarizations coming from Hodge theory, and these are known to be non-degenerate.\N\NThe authors give the following application:\N\NCorollary: Let \(X\) be a hyper-Kähler variety, and let \(f\colon X\to B\) be a Lagrangian fibration with integral fibers. Then all perverse sheaves occuring in the decomposition theorem for \(f\) have dense support.\N\NThe paper is both concise and well-written, making it a real pleasure to read.