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A note on compact solvmanifolds with Kähler structures. (English) Zbl 1105.32017
In the paper [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 D. Arapura and 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 [C. Benson and C. S. Gordon, Proc. Am. Math. Soc. 108, 971–980 (1990; Zbl 0689.53036)].

MSC:
32Q15 Kähler manifolds
32M10 Homogeneous complex manifolds
53C30 Differential geometry of homogeneous manifolds
53D05 Symplectic manifolds (general theory)
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