Resumen de Vector Fields and the Cohomology Ring of Toric Varieties

Kiumars Kaveh

• Let X be a smooth complex projective variety with a holomorphic vector field with isolated zero set Z. From the results of Carrell and Lieberman there exists a filtration F0 subset F1 subset ... of A(Z), the ring of mathbb C-valued functions on Z, such that Gr A(Z) \cong H*(X, mathbb C) as graded algebras. In this note, for a smooth projective toric variety and a vector field generated by the action of a 1-parameter subgroup of the torus, we work out this filtration. Our main result is an explicit connection between this filtration and the polytope algebra of X.