On the diffeomorphism types of certain algebraic surfaces. I
type: Item
instance of: scholarly article
title: On the diffeomorphism types of certain algebraic surfaces. I
zbMATH Open document ID: 0669.57016
author: John W. Morgan
author: Robert Friedman
published in: Journal of Differential Geometry
publication date: 1988-01-01T00:00:00Z
review text: [Part II is reviewed below (see Zbl 0669.57017).] \textit{S. K. Donaldson} [C. R. Acad. Sci., Paris, Sér. I, Math. 301, 317- 320 (1985; Zbl 0584.57010)] provided the first example of simply connected h-cobordant 4-manifolds which are not diffeomorphic, by showing that two well-known simply connected algebraic surfaces of type (1,9) are not diffeomorphic. Using the same invariant (introduced by Donaldson [loc. cit.]) the authors strengthen this result considerably, by providing the existence of infinitely many diffeomorphism types in the homotopy class of a rational elliptic surface. In particular, they study certain minimal elliptic surfaces S(p,q) which arise from rational elliptic surfaces via logarithmic transformations of coprime indices p and q. The paper contains several other important results, mostly concerning the surfaces S(p,q). An exposition of this work can be found in a survey by the authors [Algebraic surfaces and 4-manifolds: some conjectures and speculations, Bull. Am. Math. Soc., New Ser. 18, No.1, 1- 19 (1988; Zbl 0662.57016)].
Mathematics Subject Classification ID: 57R55
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 14J15
Mathematics Subject Classification ID: 32G05
Mathematics Subject Classification ID: 32J15
zbMATH DE Number: 4096396
zbMATH Keywords: smooth 4-manifolds
zbMATH Keywords: Donaldson's invariant
zbMATH Keywords: moduli space of holomorphic vector bundles
zbMATH Keywords: infinitely many diffeomorphism types in the homotopy class of a rational elliptic surface
zbMATH Keywords: minimal elliptic surfaces
Wikidata QID: Q115183915
reviewed by: Dušan Repovš
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214441784
OpenAlex ID: W1576241471
DOI: 10.4310/JDG/1214441784
description: scientific article; zbMATH DE number 4096396