Surfaces de stein associ\'ees aux surfaces de kato interm\'ediaires

TL;DR

This paper proves that the universal cover of an intermediate Kato surface minus the preimage of its rational curves is Stein and provides a criterion for its tangent bundle to be holomorphically trivial.

Contribution

It establishes the Stein property for the universal cover minus rational curves and characterizes when its tangent bundle is trivial, extending known results to intermediate Kato surfaces.

Findings

The surface $ ilde{S} ackslash ilde{D}$ is Stein.

A necessary and sufficient condition for the tangent bundle to be trivial.

Extension of known results from Enoki and Inoue-Hirzebruch surfaces.

Abstract

Let $S$ be an intermediate Kato surface, $D$ the divisor consisting of all rational curves of $S$, $\widetilde{S}$ the universal covering of $S$ and $\widetilde{D}$ the preimage of $D$ in $\widetilde{S}$. We prove two results about the surface $\widetilde{S}\setminus \widetilde{D}$: it is Stein (which was already known when $S$ is either a Enoki or a Inoue-Hirzebruch surface) and we give a necessary and sufficient condition so that its holomorphic tangent bundle is holomorphically trivialisable. ----- Soient $S$ une surface de Kato interm\'ediaire, $D$ le diviseur form\'e des courbes rationnelles de $S$, $\widetilde{S}$ le rev\^etement universel de $S$ et $\widetilde{D}$ la pr\'eimage de $D$ dans $\widetilde{S}$. On donne deux r\'esultats concernant la surface $\widetilde{S}\setminus \widetilde{D}$, \`a savoir qu'elle est de Stein (ce qui \'etait connu dans le cas o\`u $S$ est une surface d'Enoki ou d'Inoue-Hirzebruch) et on donne une condition n\'ecessaire et suffisante pour que son fibr\'e tangent holomorphe soit holomorphiquement trivialisable.