TITLE: Structure of Kähler cone
QUESTION [11 upvotes]: 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.

REPLY [3 votes]: Explicit description of a Kahler cone for all hyperkahler manifolds is here:
http://arxiv.org/abs/1401.0479 (Rational curves on hyperkahler manifolds,
Ekaterina Amerik, Misha Verbitsky)