Structure of the Kähler cone

In Calabi's Extremal kahler metrics paper, MR0645743, on page 262, the author mentioned that "It is conjectured that the structure of Kähler cone is determined by a finite number of real analytic inequalities". Would someone point it out any reference for this conjecture and any known result? In particular, what kind of "real analytic inequalities" would be expected?

Any comment or reference would be appreciated.

• Calabi's expectation is not correct, in general the boundary of the Kähler cone may have countably many faces, each defined by a polynomial equation, which accumulate at some other point of the boundary (e.g. blowups of $\mathbb{P}^2$ at 10 or more very general points). But see the discussion in Lazarsfeld's book section 1.5.E for what the correct statement is, which involves the Campana-Peternell theorem and its generalization by Demailly-Paun. Jun 13, 2018 at 8:18