Vector Fields and the Cohomology Ring of Toric Varieties
Printed: Sep 2005
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 $X$.
Toric variety, torus action, cohomology ring, simple polytope, polytope algebra
14M25 - Toric varieties, Newton polyhedra [See also 52B20]
52B20 - Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]