Theorem 23.2.60DB27896B9A96

Proof. The first equality is theorem 11.6.1 applied to the smooth closed form $\kappa _\lambda $, together with (23.2.58). It expresses $\mathcal C_ X$ as an intersection of closed half-spaces, so this cone is closed.

If $a=[\mathrm d^{\mathrm c}(\gamma ^{n-1})]\in \operatorname {LG}(X)$ and $\xi =[\lambda ]\in \mathcal C_ X\setminus K_ X$, choose a closed positive current $T=\kappa _\lambda +\mathrm{dd}^{\mathrm{c}}u$. It is nonzero because it represents the nonzero class $j_ X(\xi )$. Therefore

Conversely, suppose that $a\in W_ X\setminus \operatorname {LG}(X)$. Separating the point $a$ from the nonempty open convex cone $\operatorname {LG}(X)$ in the finite-dimensional space $W_ X$ gives a nonzero linear functional $\ell $ on $W_ X$ such that

Indeed, in the usual separating inequalities the constant can be taken to be zero, by scaling the elements of the cone. By Poincaré duality, $\ell =\langle \xi ,\cdot \rangle $ for a class $\xi \in V_ X$ that is unique modulo $K_ X$, and $\xi \notin K_ X$ because $\ell \neq 0$. The first equality of the theorem gives $\xi \in \mathcal C_ X$, and then $\langle \xi ,a\rangle \leq 0$ shows that $a$ does not belong to the right side of the second equality. This proves the second equality; if $\operatorname {LG}(X)=W_ X$, there is nothing to prove. $\square$