On the automorphism group of a symplectic half-flat 6-manifold

Abstract

We prove that the automorphism group of a compact 6-manifold M endowed with a symplectic half-flat SU(3)-structure has abelian Lie algebra with dimension bounded by min{5, b1(M )}. Moreover, we study the properties of the automorphism group action and we discuss relevant examples. In particular, we provide new complete examples on T S 3 which are invariant under a cohomogeneity one action of SO(4).2010 Mathematics Subject Classification. 53C10, 57S15.

Special solutions to the Type IIA flow

“…In Proposition 3.5, we show that a compact 6-manifold with a SHF structure (ω, ψ + ) has finite automorphism group whenever the 3-form ψ + is exact. This gives new information on the automorphism group of a compact SHF manifold in addition to the results obtained in [24]. Moreover, it holds for SHF structures with J-Hermitian Ricci tensor, and suggests that the construction of new compact examples might require substantial efforts.…”

“…In the literature, these SU(3)-structures are known as symplectic half-flat (SHF for short) [6,23,24,26] or special generalized Calabi-Yau [7,8]. According to the classification by Chiossi-Salamon [4], symplectic SU(3)-structures constitute the class W − 2 ⊕ W 5 , while SHF structures determine the subclass W − 2 .…”

“…If (ω, ψ + ) is a SHF structure with J-Hermitian Ricci tensor, then the previous result implies that ψ + is an exact 3-form, as ∆ g ψ + = dd * ψ + = dw − 2 . The converse is not true, as there exists an example of SHF structure (ω, ψ + ) on S 3 × R 3 such that ψ + is exact and Ric g is not J-Hermitian, see [24]. This example is complete and invariant under the natural cohomogeneity one action of SO(4) on S 3 × R 3 .…”

“…On the other hand, there exist examples of SHF structures that are not special according to Definition 3.6. These examples include the family of SHF structures on the 6-torus constructed in [8], the complete cohomogeneity one SHF structure on S 3 × R 3 obtained in [24], and the SHF structure on the Lie algebra g 0,− 1 6,54 given in [26,Ex. 3.1].…”

“…In this last case, the examples are locally homogeneous. Further non-compact homogeneous and cohomogeneity one examples are given in [23,24]. More details can be found in Section 3.…”