Let X be the three-dimensional, complete, nonsingular, complex torus embedding corresponding to a fan and let V be the real part of X (for definitions see [1] or [3]). The aim of this note is to give a simple combinatorial formula for calculating the Betti numbers of V.
We study the cohomology ring of the Grassmannian G of isotropic n-subspaces of a complex 2m-dimensional vector space, endowed with a nondegenerate orthogonal form (here 1 ≤ n < m). We state and prove a formula giving the Schubert class decomposition of the cohomology products in H*(G) of general Schubert classes by "special Schubert classes", i.e. the Chern classes of the dual of the tautological vector bundle of rank n on G. We discuss some related properties of reduced decompositions of "barred...
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