Automorphism groups of rational elliptic surfaces with section and constant J-map

In this paper, the automorphism groups of relatively minimal rational elliptic surfaces with section which have constant J-maps are classified. The ground field is ℂ. The automorphism group of such a surface β: B → ℙ1, denoted by Au t(B), consists of all biholomorphic maps on the complex manifold B. The group Au t(B) is isomorphic to the semi-direct product MW(B) ⋊ Aut σ (B) of the Mordell-Weil groupMW(B) (the group of sections of B), and the subgroup Aut σ (B) of the automorphisms preserving a fixed section σ of B which is called the zero section on B. The Mordell-Weil group MW(B) is determined by the configuration of singular fibers on the elliptic surface B due to Oguiso and Shioda [9]. In this work, the subgroup Aut σ (B) is determined with respect to the configuration of singular fibers of B. Together with a previous paper [4] where the case with non-constant J-maps was considered, this completes the classification of automorphism groups of relatively minimal rational elliptic surfaces with section.

RESIDUES FOR SINGULAR PAIRS AND DYNAMICS OF BIHOLOMORPHIC MAPS OF SINGULAR SURFACES

We prove the existence of a parabolic curve for a germ of biholomorphic map tangent to the identity at an isolated singular point of a surface under some conditions. For this purpose, we present a Camacho–Sad type index theorem for fixed curves of biholomorphic maps of singular surfaces and develop a local intersection theory of curves in singular surfaces from an analytic approach by means of Grothendieck residues.

The closing lemma for holomorphic maps

The closing lemma is proved for various classes of holomorphic self-maps on ${\Bbb C}^k$. These classes include those consisting of biholomorphic maps, endomorphisms and symplectomorphisms.