Exotic surfaces

We construct the first exotic $\mathbb{C}{\mathbb{P}}^2\mathrm{\#}4\overline{\mathbb{C}{\mathbb{P}}^2}$ by means of rational blowdowns. Similarly, we construct the first exotic $3\mathbb{C}{\mathbb{P}}^2\mathrm{\#}{b}^{-}\overline{\mathbb{C}{\mathbb{P}}^2}$ for $b^{-}=9,8,7$ . All of them are minimal and symplectic, as they are produced from projective surfaces $W$ with Wahl singularities, and $K_W$ big and nef. In more generality, we elaborate on the problem of finding exotic $(2\chi ({\mathcal{O}}_W)-1)\mathbb{C}{\mathbb{P}}^2\mathrm{\#}(10\chi ({\mathcal{O}}_W)-K_W^2-1)\overline{\mathbb{C}{\mathbb{P}}^2}$ from these Kollár–Shepherd-Barron–Alexeev surfaces $W$, obtaining explicit geometric obstructions.