Deformations of the holomorphically parallelizable Nakamura manifold admit balanced metrics but no strong Kähler with torsion metrics when the complex structure is non-left-invariant; similar existence questions are studied for Iwasawa deformations and a 4D solvmanifold.
Hasegawa,Small deformations and non-left-invariant complex structures on six-dimensional compact solvmanifolds, Differential Geom
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Holomorphically parallelizable solvmanifolds with special metrics and their deformations
Deformations of the holomorphically parallelizable Nakamura manifold admit balanced metrics but no strong Kähler with torsion metrics when the complex structure is non-left-invariant; similar existence questions are studied for Iwasawa deformations and a 4D solvmanifold.