Standard Conjectures for the Arithmetic Grassmannian G(2, N) and Racah Polynomials

We prove the arithmetic Hodge index and hard Lefschetz conjectures for the Grassmannian G = G(2, N) parametrizing lines in projective space, for the natural arithmetic Lefschetz operator defined via the Plücker embedding of G in projective space. The analysis of the Hodge index inequality involves estimates on values of certain Racah polynomials. 0. Introduction Let X be an arithmetic variety, by which we mean a regular, projective and flat scheme over SpecZ, of absolute dimension d + 1. Assume that M = (M, ‖ · ‖) is a hermitian line bundle on X which is arithmetically ample, in the sense of [Z] and [So, §5.2]. For each p > 0 the line bundle M defines an arithmetic Lefschetz operator L̂ : ĈH p (X)R −→ ĈH p+1 (X)R α 7−→ α · ĉ1(M). Here ĈH ∗ (X)R is the real arithmetic Chow ring of [GS] and ĉ1(M) is the arithmetic first Chern class of M . In this setting, Gillet and Soulé [GS] proposed arithmetic analogues of Grothendieck’s standard conjectures [Gr] on algebraic cycles. A more precise version of the conjectures was formulated in [So, §5.3]; assuming 2p 6 d+ 1, the statement is Conjecture 1. (a) (Hard Lefschetz) The map L̂d+1−2p : ĈH p (X)R −→ ĈH d+1−p (X)R is an isomorphism; (b) (Hodge index) If the nonzero x ∈ ĈH p (X)R satisfies L̂ d+2−2p(x) = 0, then (−1) d̂eg(x L̂d+1−2p(x)) > 0. Date: December 12, 200