Topics in Enumerative Algebraic Geometry Lecture 11
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چکیده
we deduce that H(X) = Z[x0, · · · , xr]/(σ1, · · · , σr+1), where σi is the i th elementary symmetric function of r + 1 variables. In particular it follows that H(X) ≃ Z (because Σxi = 0). Remark. Let us consider the case of general G now. According to the BorelBott-Weil theorem there is one-to-one correspondence between irreducible highest weight representations of the Lie algebra g of G and line bundles on X = G/B. Let us recall this correspondence. Suppose we are given a representation of g or, equivalently, a character χ of the maximal torus of G. There is a natural projection from B to the maximal torus (C) , so we can pullback χ to B. Let Cχ be onedimensional representation of B with character χ. Then we can construct the following line bundle: G×B Cχ
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Topics in Enumerative Algebraic Geometry Lecture 19
T : C −−−−→ R+ yM integer matrix T r ←−−−− R Then the toric manifold X is defined to be J(ω)/T r , where ω is a point in K, an open cone in R (please refer to previous lectures). Assume that X is smooth, i. e. T r action on J(ω) is free, we have: H∗(X) = H∗ T r(J (ω)) We first notice that J(ω) is T-equivariantly homotopic to J(K) where J(K) = C \ ∪ (Coordinate subspaces which miss K under J) .
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