Symmetric and skew-symmetric complex structures
Giovanni Bazzoni, Alejandro Gil-García, Adela Latorre
TL;DR
The paper investigates the interplay between complex, symplectic, and pseudo-Rähler-type structures on Lie algebras, focusing on symmetric and skew-symmetric compatibility with a fixed complex structure. It classifies 4D complex-symplectic Lie algebras, and provides a constructive framework to obtain hypersymplectic structures from pairs of symplectic forms, using a recursion operator and a complex-product viewpoint. The authors prove a criterion linking hypersymplectic metrics to a pair of symplectic forms via $E=oldsymbol{eta}_{pK}^{-1}oldsymbol{eta}_{cs}$ and $g=-oldsymbol{eta}_{pK}ig(Jullet,ulletig)$, requiring $E$ to be an almost product structure, and they illustrate this with explicit 6D and 8D examples on nilmanifolds, including a 4-step nilpotent case. They culminate with a general existence result: for every $n eq 1$, there exist $4n$-dimensional nilmanifolds carrying two distinct hypersymplectic metric families, one flat and one non-flat, both complete, thereby enriching the landscape of hypersymplectic geometry on compact quotients and its potential physical applications.
Abstract
On a complex manifold $(M,J)$, we interpret complex symplectic and pseudo-Kähler structures as symplectic forms with respect to which $J$ is, respectively, symmetric and skew-symmetric. We classify complex symplectic structures on 4-dimensional Lie algebras. We develop a method for constructing hypersymplectic structures from the above data. This allows us to obtain an example of a hypersymplectic structure on a 4-step nilmanifold.