Cohomologies of locally conformally symplectic manifolds and solvmanifolds

Abstract

Abstract. We study the Morse-Novikov cohomology and its almostsymplectic counterpart on manifolds admitting locally conformally symplectic structures. More precisely, we introduce lcs cohomologies and we study elliptic Hodge theory, dualities, Hard Lefschetz Condition. We consider solvmanifolds and Oeljeklaus-Toma manifolds. In particular, we prove that Oeljeklaus-Toma manifolds with precisely one complex place, and under an additional arithmetic condition, satisfy the Mostow property. This holds in particular for the Inoue surface of type S 0 .

Structure of locally conformally symplectic Lie algebras and solvmanifolds

“…The Lie algebra r 1 4,´1 2 ,δ is not completely solvable, hence we can not use Corollary 5.2 to determine whether the lcs structures on the solvmanifold are exact. Notwithstanding, the validity of the Mostow condition for the Inoue surface of type S 0 is confirmed in [5] (see also [38]), hence we conclude, using Corollary 5.2, that the resulting lcs structure is not exact. 5.8. d 4 .…”

“…This condition prevents some manifolds which are lcs from being symplectic. Lcs geometry is currently an active research area, see [5,7,23,44,49].…”

Locally conformally Kähler solvmanifolds: a survey

“…For the de Rham cohomology, it is shown in [57] that (5) holds in the case of OT manifolds of type (s, 1). For the Morse-Novikov cohomology, it was shown in [12] that the isomorphism (6) holds for a special class of OT manifolds of type (s, 1), namely, those satisfying the Mostow condition. However, using the description of the Morse-Novikov cohomology of OT manifolds given in [53], Istrati and Otiman show that (6) holds for any OT manifold of type (s, 1), even if the Mostow condition does not hold.…”

Symplectic cohomologies and deformations

“…Remark 2.3. Most of the results holding for the Tseng and Yau's symplectic cohomology groups can be generalized to the locally conformally symplectic ( lcs for short) setting as done by D. Angella, A. Otiman and the first-named author in [3]. A new notion of lcs-Hard-Lefschetz condition is introduced which forces the lcs structure to be globally conformally symplectic.…”

“…Let X := Γ\G be the 6-dimensional nilmanifold with structure equations (0, 0, 0, 12, 14, 15 + 23 + 24) , and let ω be the invariant symplectic structure on X defined by ω = e 16 + e 25 − e 34 . Then there exists a symplectic deformation {ω t } t such that 3) ωt (X ; R) ⊕ H (1,1) ωt (X ; R) = H 3 dR (X ; R) . However, for t = 0 we have…”