The simple invariant and differentiable structures on the Horikawa surface (Q1908315)

The simple invariant of a closed 4-manifold is the sum over the signed points in the zero-dimensional anti-self-dual moduli space. For the Horikawa surface which is a minimal complex surface with \(c^2_1= p_g=4\), the simple invariant is computed in this paper as 4. The proof follows the method of [\textit{P. B. Kronheimer}, Duke Math. J. 64, 229-241 (1991; Zbl 0754.57015)] and uses the following model of the Horikawa surface: Take a nonsingular curve \(\Pi_2\) of genus 2 and choose a \(\mathbb{Z}_2\) action with 6 fixed points. Then the quotient \((\Pi_2 \times \Pi_2)/ \mathbb{Z}_2\) has 36 double points with respect to the diagonal action. By resolving all double points one gets the Horikawa surface, a complex surface of general type. Using the method of [\textit{R. E. Gompf} and \textit{T. S. Mrowka}, Ann. Math., II. Ser. 138, 61-111 (1993; Zbl 0805.57012)] the author shows that there exist infinitely many distinct differentiable structures on this Horikawa surface.