Complex reflection groups and K3 surfaces I

In this talk we study the Hasse principle for the family of "diagonal K3 surfaces of degree 2", given by the explicit equations: $$w^2 = A_1 x_1^6 + A_2 x_2^6 + A_3 x_3^6.$$ I will explain how many such surfaces, when ordered by their coefficients, have a Brauer-Manin obstruction to the Hasse principle. This is joint work with Damián Gvirtz and Masahiro Nakahara.

Let X be a complex algebraic K3 surface of degree 2d and with Picard number ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document}. Assume that X admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, ρ≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho \\ge 1$$\\end{document} when d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=1$$\\end{document} and ρ≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho \\ge 2$$\\end{document} when d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 2$$\\end{document}. For d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=1$$\\end{document}, the first example defined over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document} with ρ=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho =1$$\\end{document} was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kondō, also defined over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document}, can be used to realise the minimum ρ=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho =2$$\\end{document} for all d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 2$$\\end{document}. In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum ρ=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho =2$$\\end{document} for d=2,3,4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2,3,4$$\\end{document}. We also show that a nodal quartic surface can be used to realise the minimum ρ=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho =2$$\\end{document} for infinitely many different values of d. Finally, we strengthen a result of Morrison by showing that for any even lattice N of rank 1≤r≤10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le r \\le 10$$\\end{document} and signature (1,r-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1,r-1)$$\\end{document} there exists a K3 surface Y defined over R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}$$\\end{document} such that PicYC=PicY≅N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Pic}\\,}}Y_{\\mathbb {C}}={{\\,\ extrm{Pic}\\,}}Y \\cong N$$\\end{document}.

Two partial orders on a reflection group W, the codimension order and the prefix order, are together called the absolute order Abs(W) when they agree. We show that in this case the absolute order on a complex reflection group has the strong Sperner property, except possibly for the Coxeter group of type D n , for which this property is conjectural. The Sperner property had previously been established for the noncrossing partition lattice NC W [11, 13], a certain maximal interval in Abs(W), but not for the entire poset, except in the case of the symmetric group [8]. We also show that neither the codimension order nor the prefix order has the Sperner property for general complex reflection groups.

We study the geometry of some moduli spaces of twisted sheaves on K3 surfaces. In particular we introduce induced automorphisms from a K3 surface on moduli spaces of twisted sheaves on this K3 surface. As an application we prove the unirationality of moduli spaces of irreducible holomorphic symplectic manifolds of K3[2]-type admitting non-symplectic involutions with invariant lattices U(2) ⊕ D4(−1) or U(2) ⊕ E8(−2). This complements the results obtained in [43], [13], and the results from [29] about the geometry of irreducible holomorphic symplectic (IHS) four-folds constructed using the Hilbert scheme of (1, 1) conics on Verra four-folds. As a byproduct we find that IHS four-folds of K3[2]-type with Picard lattice U(2) ⊕ E8(−2) naturally contain non-nodal Enriques surfaces.

Kuga and Satake associate with every polarized complex K3 surface (X, ℒ) a complex abelian variety called the Kuga-Satake abelian variety of (X, ℒ). We use this construction to define morphisms between moduli spaces of polarized K3 surfaces with certain level structures and moduli spaces of polarized abelian varieties with level structure over ℂ. In this note we study these morphisms. We prove first that they are defined over finite extensions of ℚ. This is done by proving analogues of the main theorems of complex multiplication for abelian varieties for K3 surfaces. Then we show that they extend in positive characteristic. In this way we give an indirect construction of Kuga-Satake abelian varieties over an arbitrary base. We also give some applications of this construction to canonical lifts of ordinary K3 surfaces.

In paper [7], generalising Arnold’s approach to boundary singularities, we studied smoothings of simple hypersurfaces invariant under a unitary reflection of finite order. The reflection splits the homology with complex coefficients of a symmetric Milnor fibre into a direct sum of the character subspaces H χ. The monodromy in the space of hypersurfaces with the same symmetry preserves the splitting. It was observed in [7] that the monodromy on each of the H χ is a finite group generated by unitary reflections [13]. This way unitary reflection groups, G(m, 1, k) and seven exceptional groups, made their first appearance in singularity theory addressing one of the long-standing questions of Arnold on realisations of the Shephard-Todd groups [1].

Zonal spherical functions on the complex reflection groups and ( n+1, m+1)-hypergeometric functions

We determine all complex K3 surfaces with Picard rank 20 over ℚ. Here the Neron–Severi group has rank 20 and is generated by divisors which are defined over ℚ. Our proof uses modularity, the Artin–Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell–Weil rank 18 over ℚ is impossible for an elliptic K3 surface. We apply our methods to general singular K3 surfaces, that is, those with Neron–Severi group of rank 20, but not necessarily generated by divisors over ℚ.

We develop a mixed-characteristic version of the Mori-Mukai technique for producing rational curves on K3 surfaces. We reduce modulo p, produce rational curves on the resulting K3 surface over a finite field, and lift to characteristic zero. As an application, we prove that all complex K3 surfaces with Picard group generated by a class of degree two have an infinite number of rational curves.

We report on a new approach, as well as some related experiments, to construct families of K3 surfaces having real or complex multiplication. The approach is based on an explicit description of the transcendental part of the cohomology in a topological way, using topological tori. Fundamental ideas include considering the period space of marked K3 surfaces, determining the periods by numerical integration, as well as tracing the modular curve by a numerical continuation method.

Consider a family of K3 surfaces over a hyperbolic curve (i.e., Riemann surface). Their second cohomology groups form a local system, and we show that its top Lyapunov exponent is a rational number. One proof uses the Kuga–Satake construction, which reduces the question to Hodge structures of weight 1. A second proof uses integration by parts. The case of maximal Lyapunov exponent corresponds to modular families coming from the Kummer construction.

We study a relation between coupling introduced by Ebeling (Kodai Math J 29:319-336, 2006) and the polytope duality among families of K3 surfaces.

Families of smooth closed oriented 4-manifolds with a complex spin structure are studied by means of a family version of the Bauer-Furuta invariants. The definition is given in the context of parametrised stable homotopy theory, but an interpretation in terms of characteristic cohomotopy classes on Thom spectra associated to the classifying spaces of complex spin diffeomorphism groups is given as well. The theory is illustrated with families of K3 surfaces and mapping tori of diffeomorphisms. It is also related to equivariant invariants.

We introduce the class of projective reflection groups which includes all complex reflection groups. We show that several aspects involving the combinatorics and the representation theory of complex reflection groups find a natural description in this wider setting. On introduit la classe des groupes de réflexions projectifs, ce qui généralise la notion de groupe engendré par des réflexions. On montre que plusieurs aspects concernant la combinatoire et la théorie des représentations des groupes de réflexions complexes trouvent une description naturelle dans ce cadre plus général.

We study the number of ways of factoring elements in the complex reflection groups $G(r,s,n)$ as products of reflections. We prove a result that compares factorization numbers in $G(r,s,n)$ to those in the symmetric group $S_n$ , and we use this comparison, along with the Ekedahl, Lando, Shapiro, and Vainshtein (ELSV) formula, to deduce a polynomial structure for factorizations in $G(r,s,n)$ .