Complete algebraic vector fields on affine surfaces

Let [Formula: see text] be the subgroup of the group [Formula: see text] of holomorphic automorphisms of a normal affine algebraic surface [Formula: see text] generated by elements of flows associated with complete algebraic vector fields. Our main result is a classification of all normal affine algebraic surfaces [Formula: see text] quasi-homogeneous under [Formula: see text] in terms of the dual graphs of the boundaries [Formula: see text] of their SNC-completions [Formula: see text].

Carleman approximation by holomorphic automorphisms of ℂ n

We approximate smooth maps defined on non-compact totally real manifolds by holomorphic automorphisms of \mathbb{C}^{n} .

Logarithmic connections on principal bundles over a Riemann surface

Let [Formula: see text] be a holomorphic principal [Formula: see text]-bundle on a compact connected Riemann surface [Formula: see text], where [Formula: see text] is a connected reductive complex affine algebraic group. Fix a finite subset [Formula: see text], and for each [Formula: see text] fix [Formula: see text]. Let [Formula: see text] be a maximal torus in the group of all holomorphic automorphisms of [Formula: see text]. We give a necessary and sufficient condition for the existence of a [Formula: see text]-invariant logarithmic connection on [Formula: see text] singular over [Formula: see text] such that the residue over each [Formula: see text] is [Formula: see text]. We also give a necessary and sufficient condition for the existence of a logarithmic connection on [Formula: see text] singular over [Formula: see text] such that the residue over each [Formula: see text] is [Formula: see text], under the assumption that each [Formula: see text] is [Formula: see text]-rigid.

HOLOMORPHIC AUTOMORPHISMS AND PROPER HOLOMORPHIC SELF-MAPPINGS OF A TYPE OF GENERALISED MINIMAL BALL

In this paper, we first give a description of the holomorphic automorphism group of a convex domain which is a simple case of the so-called generalised minimal ball. As an application, we show that any proper holomorphic self-mapping on this type of domain is biholomorphic.

A NOTE ON THE CONNECTEDNESS OF THE BRANCH LOCUS OF RATIONAL MAPS

Milnor proved that the moduli space Md of rational maps of degree d ≥ 2 has a complex orbifold structure of dimension 2(d − 1). Let us denote by ${\mathcal S}$d the singular locus of Md and by ${\mathcal B}$d the branch locus, that is, the equivalence classes of rational maps with non-trivial holomorphic automorphisms. Milnor observed that we may identify M2 with ℂ2 and, within that identification, that ${\mathcal B}$2 is a cubic curve; so ${\mathcal B}$2 is connected and ${\mathcal S}$2 = ∅. If d ≥ 3, then it is well known that ${\mathcal S}$d = ${\mathcal B}$d. In this paper, we use simple arguments to prove the connectivity of ${\mathcal S}$d.