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

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.

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