Vector Fields and the Cohomology Ring of Toric Varieties

[摘要] Let $X$ be a smooth complexprojective variety with a holomorphic vector field with isolatedzero set $Z$. From the results of Carrell and Liebermanthere exists a filtration$F_0 subset F_1 subset cdots$ of $A(Z)$, the ring of$c$-valued functions on $Z$, such that $Gr A(Z) cong H^*(X,c)$ as graded algebras. In this note, for a smooth projectivetoric 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 filtrationand the polytope algebra of $X$.

[学科分类] 数学（综合）

[关键词] Toric variety;torus action;cohomology ring;simple polytope;polytope algebra