A note on compact solvmanifolds with Kähler structures (Q2504128)

In the paper [\textit{K. Hasegawa}, Geom. Dedicata 78, 253--258 (1999; Zbl 0940.53037)] the author had stated a general conjecture on compact Kähler manifolds: a compact solvmanifold admits a Kähler structure if and only if it is a finite quotient of a complex torus, which has a structure of a complex torus bundle over a complex torus. In the same paper he showed that this conjecture is valid under some restriction. Based on a result of \textit{D. Arapura} and \textit{M. Nori} [Compos. Math. 116, 173--188 (1999; Zbl 0971.14020)] in the present work it is shown that the conjecture is in fact true without any restriction. As a consequence, it follows that a compact solvmanifold of completely solvable type has a Kähler structure if and only if it is a complex torus (known as the Benson-Gordon conjecture [\textit{C. Benson} and \textit{C. S. Gordon}, Proc. Am. Math. Soc. 108, 971--980 (1990; Zbl 0689.53036)].