A complex surface of general type with 𝑝_{𝑔}=0, 𝐾²=2 and 𝐻₁=ℤ/4ℤ

Abstract

We construct a new minimal complex surface of general type with p g = 0 p_g=0 , K 2 = 2 K^2=2 and H 1 = Z / 4 Z H_1=\mathbb {Z}/4\mathbb {Z} (in fact, π 1 alg = Z / 4 Z \pi _1^{\text {alg}}=\mathbb {Z}/4\mathbb {Z} ), which settles the existence question for numerical Campedelli surfaces with all possible algebraic fundamental groups. The main techniques involved in the construction are a rational blow-down surgery and a Q \mathbb {Q} -Gorenstein smoothing theory.