Invariante Divisoren und Schnitthomologie von torischen Varietäten

In this article, we complete the interpretation of groups of classes of invariant divisors on a complex toric variety X of dimension n in terms of suitable (co-) homology groups. In [BBFK], we proved the following result (see Satz 1 below): Let ${\displaystyle ClDi{v}^{}\_\left\{C\right\}\left(X\right)}$ and ${\displaystyle ClDi{v}^{}\_\left\{W\right\}\left(X\right)}$ denote the groups of classes of invariant Cartier resp. Weil divisors on X. If X is non degenerate (i.e., not equivariantly isomorphic to the product of a toric variety and a torus of positive dimension), then the natural homomorphisms ${\displaystyle ClDi{v}^{}\_\left\{C\right\}\left(X\right)\to H^2\left(X\right)}$ and ${\displaystyle ClDi{v}^{}\_\left\{W\right\}\left(X\right)\to H_{2n-2}^{cld}\left(X\right)}$ are isomorphisms, the inclusion ${\displaystyle ClDi{v}^{}\_\left\{C\right\}\left(X\right)\hookrightarrow ClDi{v}^{}\_\left\{W\right\}\left(X\right)}$ corresponds to the Poincaré duality homomorphism ${\displaystyle P_{2n-2}}$, and we have ${\displaystyle H_{2n-1}^{cld}\left(X\right)\cong H^1\left(X\right)=0}$. For the convenience of the reader, the proof is sketched below; it supersedes the proof for the compact case given in the report [BF]. Using suitable Künneth formulæ, that yields results valid in the degenerate case. In the present article, we use the sheaf-theoretic description of the intersection homology groups ${\displaystyle I_p{H}_{•}^{cld}\left(X\right)}$, for a perversity p, to prove that there is an open invariant subset ${\displaystyle V_p}$ of X and a natural isomorphism ${\displaystyle I_p{H}_{2n-j}^{cld}\left(X\right)\cong H^j\left(V_p\right)}$ for ${\displaystyle j\leqq 2}$. In the non degenerate case, we thus obtain an identification of ${\displaystyle I_p{H}_{2n-2}^{cld}\left(X\right)}$ with ${\displaystyle ClDi{v}^{}\_\left\{p\right\}\left(X\right)}$, the group of invariant Weil divisors on X that are Cartier divisors on ${\displaystyle V_p}$, and the vanishing result ${\displaystyle I_p{H}_{2n-1}^{cld}\left(X\right)=0}$ (see Satz 2). That divisor class group admits an explicit description in terms of the fan defining the toric variety. We use these results to treat problems of invariance of the intersection homology Betti number ${\displaystyle I_p{b}_{2n-2}^{cld}}$. Moreover, we discuss the question when the homology Chern class ${\displaystyle c_{n-1}\left(X\right)}$ lies in the subgroup ${\displaystyle I_p{H}_{2n-2}^{cld}\left(X\right)}$ of ${\displaystyle H_{2n-2}^{cld}\left(X\right)}$.