The simple invariant and differentiable structures on the Horikawa surface
type: Item
instance of: scholarly article
title: The simple invariant and differentiable structures on the Horikawa surface
zbMATH Open document ID: 0869.57020
author: Y. Kametani
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1997-05-20T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 58D29
Mathematics Subject Classification ID: 57R57
zbMATH DE Number: 847740
zbMATH Keywords: exotic differentiable structures
zbMATH Keywords: simple invariant
zbMATH Keywords: 4-manifold
zbMATH Keywords: anti-self-dual moduli space
zbMATH Keywords: Horikawa surface
reviewed by: Peter W. Michor
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225460
OpenAlex ID: W2066340778
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DOI: 10.2748/TMJ/1178225460
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description: scientific article; zbMATH DE number 847740