Structure of Kähler cone Are there examples of Kähler manifolds whose Kähler cone can be described explicitly, say spanned by certain cohomology classes?  As far as I know, Hirzebruch Surface has a complete description for its Kähler cone.
 A: Flag manifolds $G/B$ are nice: the Kähler cone is the positive Weyl chamber, with edges coming from the Poincaré duals of the Schubert divisors.
A: Generalising the case of Hirzebruch surface, you can say that toric varieties admit explicit description of Kähler cone.
Also for each Fano variety its Kähler cone is polyhedral, i.e., it is spanned by a finite number of rays (but this does not mean, that the description is easy). If you leave the class of Fano varieties unpleasant things may start to happen. For example for a generic blow up of $\mathbb CP^2$ in $n\ge 10$ points the structure of Kähler cone it is still unknown (for $n<9$ we get Fano), this is related to Nagata's conjecture.
Morrison's conjecture states that for a Calabi-Yau manifold the quotient of the Kähler cone by the group of isometries of the manifold is polyhedral. The conjecture was proved only for surfaces, there is a recent very nice paper of Burt Totaro on this topic
The cone conjecture for Calabi-Yau pairs in dimension two.
A: The cone of curves of K3 surfaces is described in this and this papers.
A: Explicit description of a Kahler cone for all hyperkahler manifolds is here:
https://arxiv.org/abs/1401.0479 (Rational curves on hyperkahler manifolds,
Ekaterina Amerik, Misha Verbitsky)
A: To Hirzebruch surfaces, you can add $\mathrm{CP}^2$, its $k$-folds blow-ups, $1\leq k\leq 8$, and some irrational ruled surfaces.
Related to this question is the determination of the symplectic cone. This is now understood for rational $4$-manifolds, ruled $4$-manifolds and their blow-ups, and also for some elliptic fibrations.
There is a nice survey by Tian-Jun Li of the relations between symplectic and Kahler cones for $4$-manifolds (and complex surfaces). See arXiv:0805.2931.
