Automorphisms of Parabolic Inoue Surfaces

We determine explicitly the structure of the automorphism group of a parabolic Inoue surface. We also describe the quotients of the surface by typical cyclic subgroups of the automorphism group. 1. Statement of Results In this note we determine the automorphism group AutS of a parabolic Inoue surface S. The corresponding result for a hyperbolic Inoue surface (Inoue-Hirzebruch surface) was obtained by Pinkham [8] more than twenty years ago. But the far easier case of parabolic Inoue surfaces does not seem explicit in the literature. Any parabolic Inoue surface S with second betti number m > 0 contains a unique smooth elliptic curve E and a cyclic of rational curves C = C1 + · · · + Cm, where the self-intersection number E2 = −m, and Ci are nonsingular rational curves with C2 i = −2 when m ≥ 2, while when m = 1, C = C1 is a rational curve with a single node with C2 = 0. S contains no other curves and hence any automorphism of S leaves C and E invariant. Let Aut0S be the identity component of AutS, and Aut1S the normal subgroup of AutS of elements which leave each Ci invariant. By [4] we know that Aut0S ∼= C, the multiplicative group of nonzero complex numbers. Our purpose is thus to determine the discrete part of AutS. We shall summarize our results in Theorem 1.1, Corollary 1.1 and the ensuing Remark below. In this note μm will denote the cyclic group of order m. Theorem 1.1. Let S be a parabolic Inoue surface with second betti number m > 0. Then we have the following: 1) Aut0S (∼= C) coincides with the center of AutS. 1