Geodesic
Vector Fields and the Cohomology Ring of Toric Varieties
Canadian mathematical bulletin, Tome 48 (2005) no. 3, pp. 414-427

Voir la notice de l'article provenant de la source Cambridge University Press

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 ${{F}_{0}}\subset {{F}_{1}}\subset \cdot \cdot \cdot$ of $A\left( Z \right)$ , the ring of $\mathbb{C}$ -valued functions on $Z$ , such that $\text{Gr }A\left( Z \right)\cong {{H}^{*}}\left( X,\mathbb{C} \right)$ 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 $X$ .
DOI : 10.4153/CMB-2005-039-1
Mots-clés : 14M25, 52B20, Toric variety, torus action, cohomology ring, simple polytope, polytope algebra 
Kaveh, Kiumars. Vector Fields and the Cohomology Ring of Toric Varieties. Canadian mathematical bulletin, Tome 48 (2005) no. 3, pp. 414-427. doi: 10.4153/CMB-2005-039-1
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