We classify invariant complex structures on 6-dimensional nilmanifolds up to equivalence. As an application, the behaviour of the associated Frölicher sequence is studied as well as its relation to the existence of strongly Gauduchon metrics. We also show that the strongly Gauduchon property and the balanced property are not closed under holomorphic deformation.1 Example 5.8. Let us consider the Lie algebra h 5 with the real basis {e 1 , . . . , e 6 } described in Theorem 2.1. Let us consider the complex structure J 0,0 given by J 0,0 e 1 = −e 2 , J 0,0 e 3 = −2e 2 − e 4 , J 0,0 e 5 = −e 6 , J 0,0 e 2 = e 1 , J 0,0 e 4 = −2e 1 + e 3 , J 0,0 e 6 = e 5 .

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In this paper, the main objective is to compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional supersolvable Lie algebras. We characterise the maximal abelian subalgebras of solvable Lie algebras and study solvable Lie algebras containing an abelian subalgebra of codimension 2. Finally, we prove that nilpotent Lie algebras with an abelian subalgebra of codimension 3 contain an abelian ideal with the same dimension, provided that the characteristic of the underlying field is not two. Throughout the paper, we also give several examples to clarify some results.

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