On a Morelli type expression of cohomology classes of toric varieties

Abstract

Let $X$ be a complete $\Q$-factorial toric variety of dimension $n$ and $\del$ the fan in a lattice $N$ associated to $X$. For each cone $\sigma$ of $\del$ there corresponds an orbit closure $V(\sigma )$ of the action of complex torus on $X$. The homology classes $\{[V(\sigma )]\mid \dim \sigma =k\}$ form a set of specified generators of $H_{n-k}(X,\Q)$. It is shown that, given $α\in H_{n-k}(X,\Q)$, there is a canonical way to express $\alpha$ as a linear combination of the $[V(\sigma )]$ with coefficients in the field of rational functions of degree $0$ on the Grassmann manifold of $(n-k+1)$-planes in $N_\Q$. This generalizes Morelli's formula for $\alpha$ the $(n-k)$-th component of the Todd homology class of the variety $X$.

arXiv comment17 pages