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# Vector Fields and the Cohomology Ring of Toric Varieties

Published:2005-09-01
Printed: Sep 2005
• Kiumars Kaveh
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## Abstract

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 $F_0 \subset F_1 \subset \cdots$ 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 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 $X$.
 Keywords: Toric variety, torus action, cohomology ring, simple polytope, polytope algebra
 MSC Classifications: 14M25 - Toric varieties, Newton polyhedra [See also 52B20] 52B20 - Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]