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In the present paper we study six dimensional solvable Lie algebras withspecial emphasis on those admitting a symplectic structure. We list all thesymplectic structures that they admit and we compute their Betti numbersfinding some properties about the codimension of the nilradical. Next, weconsider the conjecture of Guan about step of nilpotency of a symplecticsolvmanifold finding that it is true for all six dimensional unimodularsolvable Lie algebras. Finally, we consider some cohomologies for symplecticmanifolds introduced by Tseng and Yau in the context of symplectic Hogde theoryand we use them to determine some six dimensional solvmanifolds for which theHard Lefschetz property holds.

Cohomological properties of unimodular six dimensional solvable Lie algebras