Exotic surfaces

TL;DR

This paper constructs the first known examples of exotic smooth structures on certain complex projective surfaces using rational blowdowns, expanding the understanding of minimal symplectic 4-manifolds with specific topological invariants.

Contribution

It introduces new explicit constructions of exotic smooth structures on complex projective surfaces via rational blowdowns, including minimal symplectic examples with specified invariants.

Findings

First exotic $ ext{CP}^2 ext{#} 4 ar{ ext{CP}}^2$ constructed

First exotic $3 ext{CP}^2 ext{#} b^- ar{ ext{CP}}^2$ for $b^-=9,8,7$

Explicit geometric obstructions for constructing exotic surfaces from Kollár--Shepherd-Barron--Alexeev surfaces

Abstract

We construct the first exotic $\mathbb C \mathbb P^2 \# 4 \overline{\mathbb C \mathbb P^2}$ by means of rational blowdowns. Similarly, we construct the first exotic $3\mathbb C \mathbb P^2 \# b^- \overline{\mathbb C \mathbb P^2}$ 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 $(2\chi(\mathcal{O}_W)-1) \mathbb C \mathbb P^2 \# (10\chi(\mathcal{O}_W)-K^2_W-1) \overline{\mathbb C \mathbb P^2}$ from these Koll\'ar--Shepherd-Barron--Alexeev surfaces $W$, obtaining explicit geometric obstructions.