On compact complex surfaces of Kähler rank one (Q2851620)

The authors show that the Kähler rank, a quantity defined for a compact complex surface, is a bimeromorphic invariant. They also give a classification of compact complex surfaces of Kähler rank one.NEWLINENEWLINEThe Kähler rank is a quantity measuring how far the surface is from being Kähler, and was introduced by \textit{R. Harvey} and \textit{B. Lawson} [Invent. Math. 74, 169--198 (1983; Zbl 0553.32008)]. The Kähler rank of a surface is valued in \(\{0, 1, 2\}\). The surface has Kähler rank \(2\) iff it is a Kähler surface. The surface has Kähler rank \(1\) iff it is not Kähler but still admits a closed (semi-) positive \((1, 1)\)-form whose zero-locus is contained in a curve.NEWLINENEWLINEFor a non-elliptic surface \(X\) of class VII with Kähler rank \(1\) and such a closed (semi-) positive \((1, 1)\)-form \(\omega\), the authors consider the value \(I=\int_Xi\gamma^{1, 0}\wedge\gamma^{0, 1}\wedge\omega\), where \(\omega=\partial \gamma^{0, 1}\) and \(\gamma^{1, 0}=\overline{\gamma^{0, 1}}\). They show that \(X\) is bimeromorphic to a Hopf surface of Kähler rank \(1\) when \(I>0\), and that \(X\) enjoys the following condition \((*)\) when \(I=0\).NEWLINENEWLINE\((*)\) There exists a positive, multiplicatively automorphic, non-constant, pluriharmonic function on a \(\mathbb{Z}\)-covering \(\overline{X}\) of \(X\).NEWLINENEWLINEFrom this theorem, the authors obtain that the Kähler rank is a bimeromorphic invariant. Since any surface \(X\) whose minimal model is a Hopf surface can not enjoy condition \((*)\), this theorem also gives a classification of compact complex surfaces of Kähler rank one. NEWLINENEWLINENEWLINENEWLINE The authors observe that condition \((*)\) implies that \(X\) admits a holomorphic foliation of a very special type. They also conjecture that condition \((*)\) implies that \(X\) is bimeromorphic to an Inoue surface. According to the final note of this article, Brunella gave a positive answer to this conjecture, and together with their result, this completes the classification of compact complex surface of Kähler rank \(1\).