Assume $X$ is smooth "simply connected" complex projective variety and $Y\subset X$ a smooth hyperplane section. ( $Y= X\cap H$, $H\subset \mathbb{P}^n$).
Let's $NE(X)$ be the cone of effective 1-cycles modulo numerical-equivalence. Let's $\mathfrak{K}_X$ be the Kahler cone.
I have couple of question on these cones.
1- If two curves $C$ and $C'$ are numerically equivalent in $X$, then are they the same in $H_2(X)$.
2- If the answer to previous question be yes then we can look at the image of $NE(X)$ in $H_2(X,\mathbb{R})$. Is this cone open, i.e. of maximal dimension $=dim =\dim H_2(X)$?
3- If two divisors $D$ and $D'$ be numerically equivalent, then are they the same in $H^2(X)$?
In general when numerically equivalent $\rightarrow$ equivalence of homology classes.
4- Assume dim $X$=4. \dim X=4$. Then by Lefschetz hyperplane theorem $dim \dim H^2(X,\mathbb{Z})= \dim H^2(Y,\mathbb{Z})$ and we obviousely have $\mathfrak{K}_X \subset \mathfrak{K}_Y$. Is it possible for them to be not equal? i.e. is it possible to have a line bundle on $X$ which is ample on $Y$ but not on $X$? (For this part you may assume $X$ is Fano)
I may add to this list later:) . P.M. I know there are two more good discussions on Kahler Kähler cone ... on mathoverflow. i.e. http://mathoverflow.net/questions/30926/structure-of-kahler-cone and http://mathoverflow.net/questions/27249/what-does-the-ample-cone-look-like
