On the Kähler Rank of Compact Complex Surfaces

Abstract

Abstract. -Harvey and Lawson introduced the Kähler rank and computed it in connection to the cone of positive exact currents of bidimension (1, 1) for many classes of compact complex surfaces. In this paper we extend these computations to the only further known class of surfaces not considered by them, that of Kato surfaces. Our main tool is the reduction to the dynamics of associated holomorphic contractions (C 2 , 0) → (C 2 , 0). Résumé (Sur le rang de Kähler des surfaces complexes compactes)Harvey et Lawson ont introduit et calculé le rang de Kähler en relation avec le cône des courants positifs fermés de bidimension (1, 1) pour beaucoup de classes de surfaces complexes compactes. Dans ce travail nous étendons ces calculs à la seule classe de surfaces connues et qui n'avait pas été considérée par eux, celle des surfaces de Kato. Notre outil principal est la réduction à la dynamique des contractions holomorphes (C 2 , 0) → (C 2 , 0) associées.Texte reçu le 23 avril

Nonalgebraic compactifications of quotients of the cylinder

“…By construction, the algebraic part of the Siu decomposition of T is identically zero. This implies that T is not only closed but even exact [Toma,Rem.8], and moreover T is dd c -exact [Toma,Prop.4] 1 . More precisely, there exists a plurisubharmonic function F on S such that T = dd c F and F • Φ = F + λ for a suitable nonzero λ ∈ R. Remark that, by elliptic regularity, F is smooth outside C. A posteriori, F will be something like − log(−H), where H is the Green function of [DlOe].…”

Positive Currents on Non-K¨ahlerian Surfaces, Ii

“…In [8] the structure of these currents was described for elliptic, Hopf and Inoue surfaces. This description was extended to the other class of known surfaces, that of Kato surfaces, in [16]. Inspired by these structural results, by phenomena appearing in the study of the Kähler rank on surfaces, [5], as well as by Brunella's postumuous paper [3], we introduce in [6] an invariant on a class of d-exact positive currents which helps us to distinguish two classes among the known non-kählerian surfaces, namely parabolic and hyperbolic surfaces, see Section 2.…”