Manifolds Which Are Complex and Symplectic But Not Kähler

Abstract

The first example of a compact manifold admitting both complex and symplectic structures but not admitting a Kähler structure is the renowned Kodaira-Thurston manifold. We review its construction and show that this paradigm is very general and is not related to the fundamental group. More specifically, we prove that the simply-connected 8-dimensional compact manifold of [17] admits both symplectic and complex structures but does not carry Kähler metrics.2010 Mathematics Subject Classification. Primary: 53D05. Secondary: 55P62.

Locally conformal symplectic nilmanifolds with no locally conformal Kähler metrics

“…We have the following result Proposition 8 (see for instance [7,30]). Let N = Γ\G be a compact 4-dimensional nilmanifold.…”

Symplectic resolution of orbifolds with homogeneous isotropy

“…This was used in [7] to give the first example of a simply-connected symplectic 8-manifold which is non-formal, as the resolution of a suitable symplectic 8-orbifold. This manifold was proved to have also a complex structure in [3].…”

Complex symplectic structures on Lie algebras

“…Any manifold carries a Riemannian metric. There exist examples of symplectic manifolds without Kähler structures [33,40], of complex manifolds without Kähler structures [12,26] and even of manifolds which are complex and symplectic but still not Kähler, see [7,8,38].…”

“…e 3 − ie 4 ϕ 3 = e 5 − ie 6 ϕ 4 = e 7 − ie8 and let {e i } 8 i=1 be the dual basis. One sees that g = (0, 0, −13 + 24, −14 − 23, 15 − 26, 16 + 25, 0, 0) .We choose ω C = ϕ 14 +ϕ 23 ; its real part ω = e 17 −e 28 +e 35 −e 46 is a symplectic form with respect to which J is symmetric.…”