James Carrell
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April 8, 2024

MATH 126

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In this expository talk, let X be a smooth complex projective variety and V a holomorphic vector field on X such that 0 < |zero(V)| < \infty. Then the cohomology algebra H*(X) over C can be recovered from local data near the zero scheme Z of V. More precisely, when the upper triangular subgroup B of SL(2,C) acts on X with exactly one fixed point and V arises from the unipotent subgroup of B, then H*(X) is exactly the coordinate ring A(Z). Here the grading is given by a natural action of the diagonal torus T in B on A(Z). Moreover, the T-equivariant cohomology H*_T(X) is obtained from the union of all B-stable curves in X x P^1 for the natural action of B on P^1. In a different vein, one can ask about singular subvarieties Y of X to which V is tangent. Here, the natural result would be that H*(Y) is isomorphic to A(Y\cap Z), where Y\cap Z is the schematic intersection of Y and Z. In fact, there is a general result in this direction which we will describe if time permits. A lovely illustration of this is given by the flag variety F(n) of GL(n,C) where, using a certain B-action, it has been shown that for any B-stable Schubert variety Y in F(n), H*(Y) is isomorphic to A(Y\cap Z).

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April 8, 2024


MATH 126

, , CA

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