Cohomological properties of unimodular six dimensional solvable Lie algebras

Abstract

In the present paper we study six dimensional solvable Lie algebras with special emphasis on those admitting a symplectic structure. We list all the symplectic structures that they admit and we compute their Betti numbers finding some properties about the codimension of the nilradical. Next, we consider the conjecture of Guan about step of nilpotency of a symplectic solvmanifold finding that it is true for all six dimensional unimodular solvable Lie algebras. Finally, we consider some cohomologies for symplectic manifolds introduced by Tseng and Yau in the context of symplectic Hogde theory and we use them to determine some six dimensional solvmanifolds for which the Hard Lefschetz property holds.

Symplectic Bott–Chern cohomology of solvmanifolds

“…In this section, we apply the results in [5] in order to provide tools for the computations of the symplectic cohomologies for solvmanifolds. In particular, we recover a theorem by M. Macrì, [40], for completely-solvable solvmanifolds, see Theorem 3.2, and we extend the result to the general case in Theorem 6.8. Such results will be used in Section 4 to investigate explicit examples.…”

“…We compute the symplectic cohomologies of the above manifolds endowed with the indicated leftinvariant symplectic structures. Note that, in case of the nilmanifolds (a), (b), and (e), by Theorem 6.8, see also [40,Theorem 3], it suffices to consider the left-invariant forms. On the other hand, since the symplectic structures on the manifolds (c) and (d) satisfy the Hard Lefschetz Condition, we know that the Bott-Chern, Aeppli, and de Rham cohomologies are all isomorphic.…”

Symplectic harmonicity and generalized coeffective cohomologies

“…A characterization of the HLC in the compact case from anà la Frölicher inequality is given in [3]. Note that for the Bott-Chern and Aeppli type symplectic cohomologies, a similar result to Proposition 3.9 is obtained in [19,Theorem 3] (see also [2, Theorem 2.31]) by using another argument.…”

“…In addition, since the Euler characteristic of a nilmanifold vanishes, we have that b 3 = 2(b 2 − b 1 + 1). Therefore, relations (16)-(18) for 6-dimensional symplectic nilmanifolds are (19) Recall that c…”

Formality and the Lefschetz property in symplectic and cosymplectic geometry

“…Symplectically aspherical manifolds play an important role in symplectic geometry, see [61] for a survey on this topic. [72]. By a recent result of Hasegawa [55], a compact solvmanifold admits a Kähler structure if and only if it is a finite quotient of a complex torus which has the structure of a complex torus bundle over a complex torus.…”