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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.

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  • $\begingroup$ 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. $\endgroup$ Apr 15 '11 at 15:07
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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.

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  • $\begingroup$ where have you got the book? :) $\endgroup$ Apr 17 '11 at 21:32
  • $\begingroup$ @Dmitri A draft of the book is freely available: cs.amherst.edu/~dac/toric.html $\endgroup$
    – user2171
    Apr 17 '11 at 22:20
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http://www.cs.amherst.edu/~dac/toric.html

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