Cycle-Level Intersection Theory for Toric Varieties

[摘要] This paper addresses the problem of constructing a cycle-level intersection theory for toric varieties. We show that by making one global choice, we can determine a cycle representativefor the intersection of an equivariant Cartier divisor with an invariant cycle on a toric variety. For a toric varietydefined by a fan in $N$, the choice consists of giving an inner product or a complete flag for $M_Q=Qt Hom(N,mathbb{Z})$, or moregenerally giving for each cone $s$ in the fan a linear subspace of $M_Q$ complementary to $s^perp$, satisfying certain compatibilityconditions. We show that these intersection cycles have properties analogous to the usual intersections modulo rational equivalence. If $X$ is simplicial (for instance, if $X$ is non-singular), we obtain a commutative ring structureto the invariant cycles of $X$ with rational coefficients. This ring structure determines cycles representing certain characteristic classes of the toric variety. We also discusshow to define intersection cycles that require no choices, at the expense of increasingthe size of the coefficient field.

[学科分类] 数学（综合）