Complex structure on the rational blowdown of sections in $E(4)$

Abstract

We show that there is a complex structure on the symplectic 4-manifold $W_{4,k}$ obtained from the elliptic surface E(4) by rationally blowing down $k$ sections for $2 \leq k \leq 9$. And we interpret it via $\mathbb{Q}$-Gorenstein smoothing. This answers affirmatively to a question raised by R. Gompf.