Question on Kähler/ample cone, cone of curves....

Question

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 $=\dim H_2(X)$?

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 $\mathfrak{K}_X \subset \mathfrak{K}_Y$. 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 Kähler cone ... on mathoverflow. i.e. Structure of Kähler cone and What does the ample cone look like?

Answer 1

EDIT Added a concrete example for the answer for Q4.

By a result of Kleiman (somewhere in SGA6 and also in Lazarsfeld's book around 1.1.20) a numerically trivial line bundle has a power that's in ${\rm Pic}^\circ X$ which is trivial if $X$ is simply connected, so on such an $X$ a numerically trivial line bundle is necessarily torsion, but that corresponds to a finite étale cover, which is again trivial by the simple connectedness assumption, so there are no numerically trivial line bundles.

The long exact cohomology sequence of the exponential sequence $$ 0\to \mathbb Z \to \mathscr O_X \to \mathscr O_X^* \to 0 $$ Shows that the kernel of the map ${\rm Pic} X\to H^2(X,\mathbb Z)$ is ${\rm Pic}^\circ X$. By the above this means that ${\rm Pic} X\to H^2(X,\mathbb Z)$ is injective and since there are no torsion line bundles, it remains injective after tensoring with $\mathbb R$.

Then the answers are:

1) Yes, if $X$ is a surface since curves are divisors. In general, without the simple connectedness assumption the answer is certainly no, but that was not what you asked.

2) Actually no. Mostly no. Many examples, perhaps the simplest one is any (smooth projective) curve with $b_2>1$ or any smooth projective surface: $NE(X)$ will land in $H^{1,1}$ but $H^{2,0}\neq 0$. But even greate\r difference is possible. Take a general K3 surface: $NE(X)$ is $1$-dimensional and $H^2$ is $22$.

3) Yes, by the first two paragraphs.

4) No. Or yes, it is possible to have a line bundle like that. Let $\pi: X\to Z$ be a flat morphism from a smooth projective $4$-fold $X$ to a smooth projective $3$-fold $Z$, for instance a $\mathbb P^1$-bundle. Let $H\subset X$ be the pull-back of an ample line bundle from $Z$. Now if $Y$ is a general hyperplane section of $X$ that does not contain any fibre of $\pi$, then the induced map $Y\to Z$ is finite and hence $H\cap Y$ is an ample divisor.

Here is a concrete example: Let $Z$ be an arbitrary smooth projective threefold and let $\mathscr E$ be a rank $2$ vector bundle on $Z$ such that there exists a surjective map $\mathscr E\to \mathscr L$ onto a line bundle on $Z$. Consider $\pi: X=\mathbb P(\mathscr E)\to Z$ and observe that after twisting by the pull-back of a sufficiently ample line bundle on $Z$ we may assume that $\mathscr O_{\mathbb P(\mathscr E)}(1)$ is very ample. The surjective morphism $\mathscr E\to \mathscr L$ produces a section of $\pi$ and let us denote the image of that with $Y$. Just for kicks, notice that $Y\simeq Z$ so it can be any smooth projective variety and $Y\cap H$ corresponds to the divisor that pulls back to $H$ so it is indeed ample on $Y$. Also notice that $Y$ is a divisor on $X$ representing $\mathscr O_{\mathbb P(\mathscr E)}(1)$, so by the above assumption it is a hyperplane section of $X$. Choosing $\mathscr E\to \mathscr L$ generally makes $Y$ general but this is actually not necessary for the construction. Since $Y$ is a section, it intersects every fiber of $\pi$ in exactly $1$ point, so it cannot contain a fibre.

---

3.5) (inspired by Dave's answer): An interesting related problem is the connection between algebraic and homological equivalence. Again algebraic implies homological equivalence, but by a result of Griffiths (On the periods of certain rational integrals. I, II. Ann. of Math. (2) 90 (1969), 460-495; (2) 90 1969 496–541) they are not the same and in fact the difference can be quite large as proved by Clemens (Homological equivalence, modulo algebraic equivalence, is not finitely generated. Inst. Hautes Études Sci. Publ. Math. No. 58 (1983), 19–38 (1984)).

Answer 2

To augment Sándor's answer, especially for "Question 3.5": On a smooth variety $X$, for cycles of any codimension, homological equivalence always implies numerical equivalence, so you're asking when the two are the same. They certainly differ by torsion, so let's use ${\Bbb Q}$ coefficients and ask when $$Z^k/Hom^k \to Z^k/Num^k$$ is an isomorphism. This question is part of Grothendieck's "standard conjectures" and is unresolved in general; however it is always true for $k=1$, and apparently also for $k=2$, as well as for any $k$ on abelian varieties; see Fulton's Intersection Theory, 19.3. (In case the notation isn't self-explanatory, $Z^k$ is codimension $k$ cycles, $Hom^k$ is the kernel of the map to $H^{2k}$, and $Num^k$ is cycles numerically equivalent to zero.)

Answer 3

This is an answer to the two more questions asked in the comments. I started it out as a comment, but got tired of the space restriction....

1. I am not certain, but you are right, $NE(X)$ for this $X$ has rank at least $2$. I think that depending on the actual cover you choose the Picard number of $X$ may or may not be larger than $2$. Here is how you can study this: I think the key is to understand the Picard group of the fibers. It seems that in the general case all of these Picard groups should have rank $1$. If that's so, then using cohomology and base change (Hartshorne, Chapter III, Section 12) you can prove that then the Picard number of $X$ is $2$. In that case $NE(X)$ can be of only one type.

If the Picard number of $X$ is larger than $2$, then it gets more complicated. For a description of what can happen for a K3 you could look at The cone of curves of a K3 surface and The Cone of Curves of K3 Surfaces Revisited. I think that if you identify the relative cone of the fibration, you have a good chance at figuring out $NE(X)$ or at least enough about it so you can do whatever you need this for. On the cone of CY manifolds there are some results by Wilson, Morrison, Kawamata, Totaro.

2. Yes, the Kähler cone is the dual of $NE(X)$ in the sense you wrote it. This follows from Kleiman's criterion for ampleness: If $D\subset X$ is a divisor on a smooth projective variety, then it is ample if and only if $D\cdot\sigma>0$ for any $\sigma\in\overline{NE}(X)\setminus\{0\}$. This can be found in pretty much any book dealing with higher dimensional (birational) geometry, for example Lazarsfeld's Positivity in Algebraic Geometry I.

Answer 4

There is an interesting special case when the answer to your Question 4 is positive. Assume that $X$ is a smooth fourfold, and $Y$ an ample anticanonical hypersurface (in particular then $X$ is Fano and $Y$ is Calabi-Yau). In this case, the Kahler cones of $X$ and $Y$ coincide. This is proved by Kollar in an appendix to Borcea, Homogeneous vector bundles and families of Calabi-Yau threefolds II, in: Proc. Symp. Pure Math. 52. Part II.