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Standard conjectures for the arithmetic Grassmannian \(G(2,N)\) and Racah polynomials. (English) Zbl 1072.14514

Summary: We prove the arithmetic Hodge index and hard Lefschetz conjectures for the Grassmannian \(G(2,N)\) parametrizing lines in projective space, for the natural arithmetic Lefschetz operator defined via the Plücker embedding in projective space. The analysis of the Hodge index inequality involves estimates on values of certain Racah polynomials.

MSC:

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14M15 Grassmannians, Schubert varieties, flag manifolds
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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