A note on compact solvmanifolds with Kahler structures

By K. Hasegawa. 2004

88 citations recorded by the source corpus.

In this note we show that a compact solvmanifold admits a Kstructure if and only if it is a finite quotient of a complex torus which has a structure of a com- plex torus bundle over a complex torus. We can show in particular that a compact solvmanifold of completely solvable type has a Kstructure if and only if it is a complex torus, which is known as the Benson-Gordon's conje cture.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7d181db124b8f4ebbecda9d65b34a7effab357ba

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In Geometry and Topology

Citations in the corpus

Vaisman metrics on solvmanifolds and Oeljeklaus–Toma manifolds (2012, background): “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 (see [12] and [5]).”
Minimal models, formality and hard Lefschetz properties of solvmanifolds with local systems (2010, background): “In [1] a proof of Hasegawa’s conjecture was also written but we notice that this proof contains a gap and Hasegawa complement in [17].”
A class of Sasakian 5-manifolds (2008, background): “Let g be a 5-dimensional Lie algebra admitting a Sasakian structure.”
Balanced Hermitian structures on almost abelian Lie algebras (2020, background): “By [38], these are all finite quotients of complex tori which have the structure of a complex torus bundle over a complex torus.”
Complex and Kähler structures on Compact Solvmanifolds (2005, background): “For the details of the proof and the related topics, we refer to the paper [21].”
A correction to “Tamed symplectic forms and strong Kähler with torsion metrics” (2012, background): “, compact quotients of solvable Lie groups by discrete subgroups, Hasegawa showed in [24, 25] that a solvmanifold has a Kähler structure if and only if it is covered by a finite quotient of a complex torus, which has the structure of a complex torus bundle over a complex torus.”
On Low-Dimensional Solvmanifolds (2009, result): “This is not correct, as first noted by Hasegawa in [33].”
A survey on strong KT structures (2009, background): “Moreover, in the case of solvmanifolds, Hasegawa proved in [27, 28] that a solvmanifold carries a Kähler metric if and only if it is covered by a finite quotient of a complex torus, which has the structure of a complex torus bundle over a complex torus.”