Abstract
The first author was partially supported by FCT (Portugal) through the sabbatical grant SFRH/BSAB/135549/2018 and through CMAT (UID/MAT/00013/2013). The second author was partially supported by CIMI - Labex Toulouse - through the grant Dynamics of modular vector fields, Dwork family, and applications. The third author was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020. Finally all the three authors benefited from CNRS (France) support through the PICS project Dynamics of Complex ODEs and Geometry.
Highlights
This paper is a first step towards a project about finding new transcendent complete vector fields and/or new open complex surfaces
The project mentioned above is to a large extent concerned with an open complex surface M equipped either with a C-action or with a C2-action having an open orbit, which constitutes a far more symmetric case
The latter situation can equivalently be formulated by saying that the surface M is equipped with a pair of commuting complete vector fields X, Y that are linearly independent at generic points
Summary
This paper is a first step towards a project about finding new transcendent complete vector fields and/or new (transcendent) open complex surfaces (or, more generally, manifolds). Forstneric’s result holds for complex manifolds on which every negative plurisubharmonic function is constant, but it does not hold for arbitrary complex manifolds This contrasts with the statement in Corollary C, valid for every complex manifold, which shows that the condition of R-completeness always has strong consequences on the holomorphic vector field Z. Another curious consequence of Theorem B is as follows. Owing to Proposition 3.1, Palais’ construction allows us to integrate the abelian Lie algebra of Theorem A to yield a global C2-action on a complex surface M that a priori is not Hausdorff.
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