MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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

The answer is yes and can be found in section 6.3 of the book : Toric varieties,(David Cox, John Little, Hal Schenck) There they prove:

Numerical equivalence = algebraic equivalence

which gives the desired result.

share|cite|improve this answer
where have you got the book? :) – Dmitri Apr 17 '11 at 21:32
@Dmitri A draft of the book is freely available: – AlB Apr 17 '11 at 22:20

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.