Let $X$ be a compact Kähler manifolds, $Y$ a compact submanifold of $X$, the map $i:Y\to X$ is an inclusion map, then we know $Y$ is a Kähler manifold, and my question is:
For any Kähler class $\omega_Y$ of $Y$, is there always a Kähler class $\omega$ of $X$, such that the restriction of $\omega$ to the submanifold $Y$ is $\omega_Y$?
or equivalently,
Let $C_X,C_Y$ be the Kähler cones of $X,Y$ respectively, then for the map $i:Y\to X$, is the pull back $i^*:C_X\to C_Y$ a surjective map?
this question is important because if we add the condition that $H^2(Y,\mathcal O)=0$, then we can alway pick a Kähler class of $X$, such that the restriction to $Y$, denoted as $\omega_Y$, satisfies: $\omega_Y\in H^2(Y,\mathbb Q)$.
I guess it is true, but I find it hard to prove it, can someone help me? or give a counter-example? Any comment is welcome, thanks!
What about this? Take $Q=\Bbb P^1\times \Bbb P^1$, the usual quadric in $\Bbb P^3$. The Kähler form on $\Bbb P^3$ restricts to $\pi_1^*\omega + \pi_2^*\omega$, where $\omega$ is the usual Kähler form on $\Bbb P^1$ and $\pi_j$ are the obvious projections. Take instead the Kähler form $2\pi_1^*\omega + \pi_2^*\omega$ on $Q$. It cannot be the restriction of a Kähler form on $\Bbb P^3$ (since every Kähler form is cohomologous to a positive scalar multiple of the usual one).