Tamed symplectic structures on compact solvmanifolds of completely solvable type.

*(English)*Zbl 1382.32014A symplectic form \(\Omega\) on a complex manifold \((M, J)\) tames \(J\) if its \((1,1)\)-part is positive. Li and Zhang proved that this condition forces a compact surface to be Kähler [T.-J. Li and W. Zhang, Commun. Anal. Geom. 17, No. 4, 651–683 (2009; Zbl 1225.53066)], and it is also known that a Moishezon manifold cannot have such an \(\Omega\), except if it is Kähler [T. Peternell, Math. Ann. 275, 653–672 (1986; Zbl 0606.32018)]. Hence it is interesting to try to find a non-Kähler compact complex manifold admitting a taming symplectic form.

A good place to look are compact homogeneous spaces \(G/\Gamma\), in particular nilmanifolds and solvmanifolds (compact quotients of nilpotent and solvable Lie groups by lattices \(\Gamma\)), by virtue of the Benson-Gordon-Hasegawa theorem [C. Benson and C. S. Gordon, Topology 27, No. 4, 513–518 (1988; Zbl 0672.53036); K. Hasegawa, Proc. Am. Math. Soc. 106, No. 1, 65–71 (1989; Zbl 0691.53040)], whereby a compact nilmanifold is Kähler iff it is a torus. K. Hasegawa [Osaka J. Math. 43, No. 1, 131–135 (2006; Zbl 1105.32017)] extended this to show that a compact solvmanifold is Kähler iff it is a complex torus bundle over a complex torus (see also [O. Baues and V. Cortés, Geom. Dedicata 122, 215–229 (2006; Zbl 1128.53043)]).

The paper [N. Enrietti et al., J. Symplectic Geom. 10, No. 2, 203–223 (2012; Zbl 1248.53070)] proved that a compact nilmanifold with an invariant almost complex structure \(J\) (i.e., left-invariant on \(G\)) is Kähler iff it has a taming \(\Omega\). The present paper pushes this to encompass compact solvmanifolds of so-called completely solvable type, meaning that the adjoint map on the Lie algebra of \(G\) has real eigenvalues. The proof is carried out by symplectic reduction and a thorough study of the Lie algebras involved. Other results of the same flavour come in [the authors et al., Tohoku Math. J. (2) 67, No. 1, 19–37 (2015; Zbl 1325.32023)].

A good place to look are compact homogeneous spaces \(G/\Gamma\), in particular nilmanifolds and solvmanifolds (compact quotients of nilpotent and solvable Lie groups by lattices \(\Gamma\)), by virtue of the Benson-Gordon-Hasegawa theorem [C. Benson and C. S. Gordon, Topology 27, No. 4, 513–518 (1988; Zbl 0672.53036); K. Hasegawa, Proc. Am. Math. Soc. 106, No. 1, 65–71 (1989; Zbl 0691.53040)], whereby a compact nilmanifold is Kähler iff it is a torus. K. Hasegawa [Osaka J. Math. 43, No. 1, 131–135 (2006; Zbl 1105.32017)] extended this to show that a compact solvmanifold is Kähler iff it is a complex torus bundle over a complex torus (see also [O. Baues and V. Cortés, Geom. Dedicata 122, 215–229 (2006; Zbl 1128.53043)]).

The paper [N. Enrietti et al., J. Symplectic Geom. 10, No. 2, 203–223 (2012; Zbl 1248.53070)] proved that a compact nilmanifold with an invariant almost complex structure \(J\) (i.e., left-invariant on \(G\)) is Kähler iff it has a taming \(\Omega\). The present paper pushes this to encompass compact solvmanifolds of so-called completely solvable type, meaning that the adjoint map on the Lie algebra of \(G\) has real eigenvalues. The proof is carried out by symplectic reduction and a thorough study of the Lie algebras involved. Other results of the same flavour come in [the authors et al., Tohoku Math. J. (2) 67, No. 1, 19–37 (2015; Zbl 1325.32023)].

Reviewer: Simon Chiossi (Niteroi)

##### MSC:

32J27 | Compact Kähler manifolds: generalizations, classification |