On the diffeomorphism groups of certain algebraic surfaces. (English) Zbl 0753.57012

Ebeling, Wolfgang; Okonek, Christian

Enseign. Math., II. Sér. 37, No. 3-4, 249-262 (1991).

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.

Reviewer: T.Lawson (New Orleans)

MSC:

57N13	Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R50	Differential topological aspects of diffeomorphisms
14J25	Special surfaces
57R95	Realizing cycles by submanifolds

Keywords:

Moishezon surface; homology classes represented by smoothly embedded 2- spheres; diffeomorphism group; intersection form; simply connected algebraic surface; complete intersection; Salvetti surface; monodromy group