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For \\Lambda, a lattice of Cartier divisors, let R_\\Lambda denote the corresponding sheaf associated to V spanned by the first Chern classes of divisors in \\Lambda. We prove that any projective, smooth variety on which the bundle R_\\Lambda splits into a direct sum of line bundles is toric. 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Wisniewski, Oskar Kedzierski","submitted_at":"2011-04-04T20:34:29Z","abstract_excerpt":"A generalized Euler sequence over a complete normal variety X is the unique extension of the trivial bundle V \\otimes O_X by the sheaf of differentials \\Omega_X, given by the inclusion of a linear space V in Ext^1(O_X,\\Omega_X). For \\Lambda, a lattice of Cartier divisors, let R_\\Lambda denote the corresponding sheaf associated to V spanned by the first Chern classes of divisors in \\Lambda. We prove that any projective, smooth variety on which the bundle R_\\Lambda splits into a direct sum of line bundles is toric. 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