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.
Kodaira,Complex Manifolds and Deformations of Complex Structures, Classics in Mathematics, Springer Berlin, Heidelberg, 2005.↑3, 5
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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.