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Neutral Calabi-Yau structures on Kodaira manifolds. (English) Zbl 1058.32018

The authors study some classes of solutions of the Einstein equations in dimension \(4n\) with signature \((2n,2n)\). These solutions are obtained by constructing neutral Calabi-Yau metrics fibered by Lagrangian tori.
Let \(G\) be a simply connected real \(2\)-step nilpotent Lie group of dimension \(4n\), with center \(Z\) of dimension \(2n\). Assume that the Lie algebra \(\mathcal{G}\) has an integrable almost complex structure which preserves the center \(\mathcal{Z}\) of \(\mathcal{G}\). A Kodaira manifold \((M,I)\) is a compact quotient \(M=\Gamma \backslash G\) by a uniform discrete subgroup, endowed with the invariant complex structure \(I\) associated to the complex structure of \(\mathcal{G}\). A neutral Kähler manifold is a complex manifold \((M^{4n},I)\) with a neutral metric \(g\) (i.e. of signature \((2n,2n)\)) such that \(g(IX,IY)=g(X,Y)\) and \(I\) is parallel with respect to the Levi Civita connection \(\nabla\). Such a manifold is Calabi-Yau if there exists \(\Phi\in \Lambda^{2n,0}T^*M\) such that \(\Phi\neq 0\) and \(\nabla \Phi=0\).
The authors show that any principal \(T^Z\)-bundle over \(T^{2n}\) admits a neutral Ricci flat metric and that Kodaira manifolds \(\Gamma \backslash G\) with a closed \(G\)-invariant non-degenerate \((1,1)\)-form admit \(T^Z\)-invariant neutral Calabi-Yau metrics with special Lagrangian tori. They present examples of \(8\)-dimensional Kodaira manifolds with neutral Calabi-Yau and hypersymplectic structures. Next they obtain the existence of neutral Calabi-Yau structures on compact quotients of cotangent bundles of some \(2\)-step nilpotent Lie groups \(G\). On the compact quotients \(T^*(H_3\times{\mathbb{R}})\) there exist families of neutral Calabi-Yau structures which contain also hypersymplectic structures.

MSC:

32Q25 Calabi-Yau theory (complex-analytic aspects)
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
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