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

TL;DR

This paper demonstrates the existence of a complex structure on certain symplectic 4-manifolds derived from E(4) through rational blowdowns, using Q-Gorenstein smoothing, answering a question by Gompf.

Contribution

It establishes the presence of complex structures on rational blowdowns of E(4) sections, linking symplectic topology with complex geometry via Q-Gorenstein smoothing.

Findings

Existence of complex structures on W_{4,k} for 2 ≤ k ≤ 9.

Interpretation of complex structures through Q-Gorenstein smoothing.

Affirmative answer to Gompf's question.

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 $2\le k\le 9$. And we interpret it via ${\mathbb Q}$-Gorenstein smoothing. This answers affirmatively to a question raised by R. Gompf.