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Construction of complex contact manifolds via reduction. (English) Zbl 1327.53104

This paper presents several results in complex contact geometry. Let \((M,J)\) be a complex manifold of dimension \(2n+1\). A complex contact structure on \((M,J)\) is given locally by a holomorphic 1-form \(\omega\) such that \(\omega \wedge (d\omega)^n \neq 0\). The holomorphic sub-bundle \(\ker \omega\) of \(TM\) is nonintegrable and is called horizontal. A complex almost contact metric structure on \((M,J)\) is given by a Hermitian metric \(g\), and local 1-forms \(u,v\), \((1,1)\)-tensors \(G,H\), and vector fields \(U,V\) satisfying certain compatibility conditions and relations \(H=GJ\), \(V=-JU\), \(GJ=-JG\), \(H^2 = G^2 = -I + u \otimes U + v \otimes V\), \(g(G \cdot, \cdot) = -g(\cdot, G\cdot)\), \(g(U,\cdot) = u(\cdot)\), \(GU=0\) and \(u(U) = 0\). As such, \((u,U,G,g)\) and \((v,V,H,g)\) resemble (but are not exactly) two real (almost) contact metric structures, related by the complex structure \(J\). A complex contact metric structure on \((M,J)\) is a complex almost contact metric structure satisfying further integrability conditions.
A notion of normality for complex (almost) contact metric structures was defined by S. Ishihara and M. Konishi [Kodai Math. J. 3, 385–396 (1980; Zbl 0455.53032)]. Given a complex contact metric structure, Ishikara and Konishi defined “torsion” tensor fields \(S\) and \(T\); normality is defined by the vanishing of \(S\) and \(T\). This normality implies that the Hermitian manifold \((M,J,g)\) is Kähler.
The authors prove an interesting property of the sectional curvatures of normal complex contact manifolds. Let \(K(X,Y)\) denote the sectional curvature of the plane spanned by two vector fields \(X,Y\). If \(X\) is a horizontal vector field, the authors show that \[ K(X,JX) + K(X,GX) + K(X,HX) = 6. \]
The headline result of the paper is that complex contact structures can be obtained by a form of reduction, in particular the reduction from a hyper-Kähler manifold by a \(\mathbb{C}^*\) action.
Given a hyper-Kähler manifold with a free proper holomorphic action of \(\mathbb{C}^*\), the authors show that the quotient \(M \backslash \mathbb{C}^*\) is canonically endowed with a normal complex almost contact metric structure.
Using this result, the authors are able to construct a new example of a normal complex almost contact metric manifold, by taking the quotient of \(\mathbb{C}^4 \backslash \{z_1 z_2 z_3 z_4 = 0 \}\) under the \(\mathbb{C}^*\) action given by \(\lambda \cdot (z_1, z_2, z_3, z_4) = (\lambda z_1, \lambda z_2, \lambda^{-1} z_3, \lambda^{-1} z_4)\).
The authors also construct a non-normal complex almost contact metric structure, explicitly defining a complex almost contact metric structure on the product of spheres \(S^{4m+3} \times S^{4n+3}\). This structure is induced from 3-Sasakian structures on \(S^{4m+3}\) and \(S^{4n+3}\), which arise from \(S^{4n+3}\) lying as a hypersurface in the quaternionic space \(\mathbb{H}^{n+1}\). As \(H^2 (S^{4m+3} \times S^{4n+3}) = 0\), this manifold admits no Kähler structure, hence no normal complex almost contact metric structure.

MSC:

53D10 Contact manifolds (general theory)
53D15 Almost contact and almost symplectic manifolds

Citations:

Zbl 0455.53032
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Full Text: DOI Euclid

References:

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