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A note on infinitesimal deformations of symplectic half-flat structures. (English) Zbl 1179.53035

Let \(M\) be a 6-dimensional smooth manifold. A half-flat structure on \(M\) is an \(\text{SU(3)}\)-structure whose defining forms \((\omega, \psi_+)\in\wedge^2M\oplus\wedge^3M\) satisfy \(d\omega\wedge\omega=0\) and \(d\psi_+=0\). A half-flat structure \((\omega, \psi_+)\) is symplectic half-flat if \(\omega\) is a symplectic form. A symplectic form \(\omega\) satisfies the hard Lefschetz condition if the map \(\wedge\omega^p:\wedge^{3-p}M\to\wedge^{3+p}M\), defined by \(\gamma\mapsto\omega^p\wedge\gamma\), induces an isomorphism in cohomology for \(p=1,2,3\).
It is shown in this work that the moduli space of infinitesimal deformations of symplectic half-flat structure satisfying the hard Lefschetz condition has finite dimension. The technics used are the ones introduced by R. Goto in [Int. J. Math. 15, No. 3, 211–257 (2004; Zbl 1046.58002), arXiv:math/0108002].

MSC:

53C10 \(G\)-structures
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53D05 Symplectic manifolds (general theory)
32G07 Deformations of special (e.g., CR) structures

Citations:

Zbl 1046.58002
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