${{\omega}}_{{\mathrm{FS}}}$.) Now use the construction in [@MS1 Lemma 7.15] to obtain a symplectic form $\rho_0$ on $M$ such that $\rho_0=\pi^*{{\omega}}_0$ on $M\setminus\pi^{-1}(U_1\cup U_2)$ and the exceptional spheres $S_1,S_2$ are symplectic submanifolds of small areas.

Let ${{\omega}}$ be any symplectic form on $M$ with first Chern class $c:=c_1({{\omega}})$ and choose $E^c_1,E^c_2,E^c_3$ as in part 1 above such that  holds. Denote $$L^c := E^c_1+E^c_2+E^c_3,\qquad
\lambda_i := {\langle [{{\omega}}], E^c_i\rangle},\qquad i=1,2,3,$$ and assume ${\langle [{{\omega}}], L^c\rangle}=\pi$. By Taubes–Seiberg–Witten theory [@TAUBES5], the classes $E^c_1,E^c_2,L^c$ are represented by pairwise disjoint symplectically embedded spheres $C_1,C_2,C$. Hence there exist disjoint Weinstein neighborhoods $N_1,N_2\subset M\setminus C$ of $C_1,C_2$ in which the symplectic form is standard (see [@MS1 Theorem 3.30]). The symplectic blowdown construction replaces $C_1,C_2$ by closed four-balls of radii $r_1,r_2$ with $\pi r_i^2=\lambda_i$. This results in a symplectic manifold $(\widehat{M},\widehat{{{\omega}}})$, equipped with symplectic embeddings $\widehat{\psi}_i:B^4_{r_i+\delta}\to\widehat{M}$ of closed four-balls with images $\widehat{N}_i:=\widehat{\psi}_i(B^4_{r_i+\delta})$, and a symplectomorphism $$\widehat{\pi}:(M\setminus(N_1\cup N_2),{{\omega}})\to 
(\widehat{M}\setminus(\widehat{N}_1\cup\widehat{N}_2),\widehat{{{\omega}}}).$$ Thus $C$ descends to a symplectically embedded sphere $\widehat{C}:=\widehat{\pi}(C)\subset\widehat{M}$, disjoint from $\widehat{N}_1\cup\widehat{N}_2$, with area $\pi$ and self-intersection number $\widehat{C}\cdot\widehat{C}=1$. The homology class of $\widehat{C}$ generates $H_2(\widehat{M};{{\mathbb Z}})$. Hence a theorem of Gromov [@GROMOV1] and McDuff [@MCDUFF1] (see also [@MS2 Theorem 9.4.1]) asserts that there exists a symplectomorphism $g:(\widehat{M},\widehat{C},\widehat{{{\omega}}})\to({{{{\mathbb C}}\mathrm{P}}}^2,{{{{\mathbb C}}\mathrm{P}}}^1,{{\omega}}_0)$. 