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Question on Kahler\ampleKähler/ample cone, cone of curves....

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$is Fano) I may add to this list later:) . P.M. I know there are two more good discussions on Kahler Kähler cone ... on mathoverflow. i.e. http://mathoverflow.net/questions/30926/structure-of-kahler-cone and http://mathoverflow.net/questions/27249/what-does-the-ample-cone-look-like 2 added 33 characters in body Assume$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)$be the cone of effective 1-cycles modulo numerical-equivalence. Let's$K_X$\mathfrak{K}_X$ 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 $=dim H_2(X)$?

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. Then by Lefschetz hyperplane theorem $dim H^2(X,\mathbb{Z})= dim H^2(Y,\mathbb{Z})$ and we obviousely have $K_X \mathfrak{K}_X \subset K_Y$. \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$is Fano) I may add to this list later:) . P.M. I know there are two more good discussions on Kahler cone ... on mathoverflow. i.e. http://mathoverflow.net/questions/30926/structure-of-kahler-cone and http://mathoverflow.net/questions/27249/what-does-the-ample-cone-look-like 1 Question on Kahler\ample cone, cone of curves.... Assume$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)$be the cone of effective 1-cycles modulo numerical-equivalence. Let's$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 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$=4. Then by Lefschetz hyperplane theorem$dim H^2(X,\mathbb{Z})= dim H^2(Y,\mathbb{Z})$and we obviousely have$K_X \subset 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 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