# On homology of Toric varieties

Lets $X$ be a simply connected projective toric variety of dimension $n$.

Lets $\tau_1,\cdots,\tau_k$ be the set of $(n-1)$-dimensional cones of corresponding fan which is in one-to-one correspondence with invariant curves on $X$.

Both $H_2(X,\mathbb{Q})$ and $N_1(X,\mathbb{Q})$ are quotients of the vector space $\sum \mathbb{Q} \cdot \tau_i$ with respect to different equivalence relations. ( Please correct me if I am wrong) and we have a surjective map $H_2 \rightarrow N_1$.

My question is whether this is an isomorphism?

Dual to this picture we have $H^2(X) \cong Pic(X)$ and therefore a surjective map $H^2(X,\mathbb{Q}) \rightarrow N^1(X,\mathbb{Q})$; which I am asking again whether this is an isomorphism.

For those who don't know: $N^1(X)=$ formal linear sum of Cartier-divisors modulo numerical equivalence and $N_1(X)=$ formal linear sum of curves modulo numerical equivalence; Where numerical equivalence is given by intersection theory between Cartier divisors and curves.

-
Note that $N^1=Pic/Pic^0$, so the questions is: Is there a non-trivial line bundle in $Pic^0(X)$ when $X$ is toric. –  Mohammad F. Tehrani Apr 15 '11 at 15:07