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Cohomology of large semiprojective hyperkaehler varieties. (English. French summary) Zbl 1348.53058

Bost, Jean-Benoît (ed.) et al., De la géométrie algébrique aux formes automorphes (II). Une collection d’articles en l’honneur du soixantième anniversaire de Gérard Laumon. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-806-0/pbk). Astérisque 370, 113-156 (2015).
In this paper, issued on the occasion of G. Laumon’s 60th birthday, the authors use geometric and arithmetic techniques to study the cohomology (and in particular Betti numbers) of semiprojective hyper-Kähler varieties, such as toric hyper-Kähler varieties, Nakajima quiver varieties, or moduli spaces of Higgs bundles.
A quasi-projective algebraic variety \(X\) with a \(\mathbb C^{\times}\) action is called semiprojective [C. Simpson, Proc. Symp. Pure Math. 62, 217–281 (1997; Zbl 0914.14003)] if 1. the set \(X^{\mathbb C^{\times}}\) of fixed points of the action is proper, i.e., projective, and 2. for all \(x\in X\), the limit \(\displaystyle\lim_{\lambda\to 0}\lambda\cdot x\) exists. Such a variety can often be realised as the total space of a vector bundle on a projective variety with canonical \(\mathbb C^{\times}\)-action. Another good source for this are GIT quotients [T. Hausel and B. Sturmfels, Doc. Math., J. DMV 7, 495–534 (2002; Zbl 1029.53054). For example, in the hyper-Kähler case, one is brought to consider algebraic symplectic quotients of a complex symplectic vector space \(\mathbb{M}\) by a symplectic linear action, with corresponding notation \(\mathcal{M}^{\rho}_{\sigma}\) (in practice, \(\mathbb{M}= \mathbb{V}\times \mathbb{V}^*\), the representation is a doubling of a representation \(\rho:G\to \mathcal{G}\ell(\mathbb{V})\), and one also chooses a character \(\sigma\) of \(G\) to perform the quotient).
Thanks to the Morse stratification, which is perfect in the sense of Atiyah-Bott, the cohomology of \(X\) turns out to be that of \(X^{\mathbb C^{\times}}\) (with cohomological shifts). Thanks to another, somehow conjugate, stratification, the Bialynicki-Birula decomposition [C. Simpson, Proc. Symp. Pure Math. 62, 217–281 (1997; Zbl 0914.14003); H. Nakajima, Duke Math. J. 91, No. 3, 515–560 (1998; Zbl 0970.17017); M. F. Atiyah and R. Bott, Philos. Trans. R. Soc. Lond., Ser. A 308, 523–615 (1983; Zbl 0509.14014)], and the related key-notion of core, one sees that \(H^*(X, \mathbb C)\) is pure. Both these facts on cohomology are used in a “Weak” hard Lefschetz theorem (Theorem 1.4.1) stated for semiprojective varieties with equidimensional core, and called this way because of its (a priori) limited range of validity.
The purity of \(H^*(X, \mathbb C)\) is moreover heavily exploited in §2, as it makes possible, together with Katz’s theorem [the authors, Invent. Math. 174, No. 3, 555–624 (2008; Zbl 1213.14020)], the arithmetic harmonic analysis, developed previously by the first author [Proc. Natl. Acad. Sci. USA 103, No. 16, 6120–6124 (2006; Zbl 1160.53374)], on “strongly-polynomial count” complex varieties, such as the ones under study here. This technique consists in computing Betti numbers of complex varieties, by counting points in their finite-field versions (immediately available here by the GIT quotient process); this is explicitly worked out for toric hyper-Kähler (via Tutte polynomials [W. T. Tutte, Can. J. Math. 6, 80–91 (1954; Zbl 0055.17101); F. Ardila, Pac. J. Math. 230, No. 1, 1–26 (2007; Zbl 1152.52011)], Nakajima quiver varieties, twisted ADHM spaces as well as for Hilbert scheme of points of \((\mathbb C^2)^{[n]}\) (with the help of computational results from [the first author, Proc. Natl. Acad. Sci. USA 103, No. 16, 6120–6124 (2006; Zbl 1160.53374); H. Nakajima and K. Yoshioka, in: Algebraic structures and moduli spaces. Proceedings of the CRM workshop, Montréal, Canada, July 14-20, 2003. Providence, RI: American Mathematical Society (AMS). 31–101 (2004; Zbl 1080.14016); L. Göttsche, Manuscr. Math. 66, No. 3, 253–259 (1990; Zbl 0714.14004); G. Ellingsrud and S. A. Strømme, Invent. Math. 87, 343–352 (1987; Zbl 0625.14002)]), and leads to a numerical conjecture in the case of Higgs bundles moduli spaces.
In order to test a possible larger range of application for Theorem 1.4.1, the authors then concentrate on Betti number graphs of high-dimensional semiprojective varieties in the rest of the paper; for example, the toric quiver variety \(\mathcal{M}_1^{K_{40}}\) has real dimension 2964. They observe patterns such as the convergence, as dimension increases, of the Betti number graphs towards graphs of special continuous functions. For instance, for the \(\mathcal{M}_1^{K_{n}}\) family (resp. the \((\mathbb C^2)^{[n]}\) family) of examples, the authors prove the convergence to the graph of the Airy distribution (resp. the Grumbel distribution); a convergence to the graph of the Airy distribution graph in the case of large quiver varieties should also hold. The paper is concluded by a prospective discussion in the lines of these observations.
For the entire collection see [Zbl 1319.14004].

MSC:

53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14L30 Group actions on varieties or schemes (quotients)
20G05 Representation theory for linear algebraic groups
14D20 Algebraic moduli problems, moduli of vector bundles
05E05 Symmetric functions and generalizations
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