condition  is necessary for the existence of ${{\omega}}$. The converse follows from the Nakai–Moishezon criterion. (See Demazure [@DEMAZURE] and also McDuff–Polterovich [@MP] for a direct symplectic proof.)

[**4.**]{} Let $a\in H^2(M;{{\mathbb R}})$ such that $a^2>0$. Close examination shows that there exists a cohomology class $c\in{{\mathcal C}}$ such that condition  holds if and only if ${\langle a, E\rangle}\ne0$ for [*all*]{} $E\in{{\mathcal E}}$. (See Li–Liu [@LL1; @LL2] for details.)

Assertions (I), (II), (III) follow from 1.-4.  and (IV) follows from Corollary A.

![[The symplectic chamber structure on ${{{{\mathbb C}}\mathrm{P}}}^2\#2\overline{{{{{\mathbb C}}\mathrm{P}}}}^2$.]{}[]{data-label="fig:CP22"}](CP22.eps)

[**5.**]{} To understand (IV) geometrically, it is convenient to slightly change the point of view. Fix a cohomology class $c\in{{\mathcal C}}$ and define $$\label{eq:sympcone}
\begin{split}
{{\mathcal E}}^c 
&:= \left\{E\in H_2(M;{{\mathbb Z}})\,|\,
E\cdot E=-1,\,{\langle c, E\rangle}=1\right\},\\
{{\mathcal K}}^c
&:= 
\left\{a\in H^2(M;{{\mathbb R}})\,|\,a^2>0,\,
{\langle a, E\rangle}>0\;\forall\,E\in{{\mathcal E}}^c\right\},\\
{{\mathcal S}}^c 
&:= 
\left\{[\rho]\in H^2(M;{{\mathbb R}})\,|\,\rho\in{{\Omega}}^2(M),\,
d\rho=0,\,\rho\wedge\rho>0,\,c_1(\rho)=c\right\}.
\end{split}$$ Then ${{\mathcal S}}^c={{\mathcal K}}^c$ (see 3. above). These cones define a chamber structure on $H^2(M;{{\mathbb R}})$. (See Figure \[fig:CP22\], where the standard basis of $H_2(M;{{\mathbb Z}})$ is denoted by $L,E,F$ and the symplectic cohomology classes satisfy the normalization conditions ${\langle a, L\rangle}=1$, ${\langle a, E\rangle}>0$, ${\langle a, F\rangle}>0$.) The chamber structure is determined by straight lines connecting pairs of 