On the diffeomorphism groups of certain algebraic surfaces
type: Item
instance of: scholarly article
title: On the diffeomorphism groups of certain algebraic surfaces
zbMATH Open document ID: 0753.57012
author: Wolfgang Ebeling
author: Christian Okonek
published in: L'Enseignement Mathématique. 2e Série
publication date: 1992-09-26T00:00:00Z
review text: This paper studies the image of the diffeomorphism group \(\text{Diff}_ +(X)\) in the isometry group \(O(L)\) for a simply connected algebraic surface \(X\). Here, \(L\) is the lattice (\(H_ 2(X,\mathbb{Z}),q)\) where \(q\) is the intersection form. The main result of the paper completely determines \(\psi(\text{Diff}_ +(X))\) as \(O'_ k(L)\cdot\{\sigma_ *,id\}\) when \(X\) is a simply connected algebraic surface with \(p_ g(X)\equiv 1\bmod 2\) and \(k^ 2_ X\equiv 1{}\bmod 2\) which is either a complete intersection or a Moishezon or Salvetti surface. This result follows from other results which first characterize which automorphisms arise from the monodromy group, show that the canonical class is left invariant when there is a big monodromy group, and show that \(-1\) is not in \(\psi(\text{Diff}_ +(X))\) when \(p_ g(X)\equiv 1\bmod 2\). The results are applied to give information on when certain homology classes can or cannot be represented by smoothly embedded 2-spheres as well as characterize when the images are conjugate subgroups in terms of the divisibilities of the canonical classes for homeomorphic algebraic surfaces as above.
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 57R95
zbMATH DE Number: 55793
zbMATH Keywords: Moishezon surface
zbMATH Keywords: homology classes represented by smoothly embedded 2- spheres
zbMATH Keywords: diffeomorphism group
zbMATH Keywords: intersection form
zbMATH Keywords: simply connected algebraic surface
zbMATH Keywords: complete intersection
zbMATH Keywords: Salvetti surface
zbMATH Keywords: monodromy group
reviewed by: Terry Lawson
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 55793