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SKT and tamed symplectic structures on solvmanifolds. (English) Zbl 1325.32023
The paper investigates compact homogeneous manifolds $$G/\Gamma$$ equipped with invariant complex structures $$J$$ that are tamed by symplectic forms $$\Omega$$ (meaning $$\Omega^{(1,1)} > 0$$). Taming forms provide static solutions of the so-called pluriclosed flow [J. Streets and G. Tian, Int. Math. Res. Not. 2010, No. 16, 3101–3133 (2010; Zbl 1198.53077)]. Although compact complex surfaces admitting a taming form (sometimes called ‘Hermitian-symplectic’ structure) are Kähler, and Moishezon complex structures on compact manifolds cannot be tamed, it is unknown whether there exist compact Hermitian-symplectic manifolds with no Kähler structure.
More generally, one could consider strong Kähler metrics with torsion (SKT), for which the fundamental form is $$\partial\overline{\partial}$$-closed, see [N. Enrietti et al., J. Symplectic Geom. 10, No. 2, 203–223 (2012; Zbl 1248.53070)] and [A. Fino and A. Tomassini, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 52(100), No. 2, 99–116 (2009; Zbl 1199.53138)].

Here the focus is on solvmanifolds $$G/\Gamma$$, i.e., compact quotients of simply connected solvable Lie groups $$G$$ by lattices $$\Gamma$$. Whilst the picture for nilmanifolds ($$G$$ nilpotent) is pretty clear, as only tori can be tamed by symplectic forms (Enrietts et al. [loc.cit.]), little is known about genuine SKT solvmanifolds. K. Hasegawa [Osaka J. Math. 43, No. 1, 131–135 (2006; Zbl 1105.32017)] proved that a Kähler solvmanifold $$G/\Gamma$$ must essentially be a finite quotient of a torus, implying that $$G/\Gamma$$ cannot be Kähler when $$G$$ is not of type (I) and not Abelian. Type (I) means that the adjoint map ad has purely imaginary spectrum.
Recall that one calls $$G$$ a splitting Lie group whenever:
1) its Lie algebra $$\mathfrak{g}$$ is a semidirect product $$\mathfrak{g} = \mathfrak{s} \times_\phi \mathfrak{h}$$, with $$\mathfrak{s}$$ solvable, $$\mathfrak{h}$$ a Lie algebra;
2) $$\phi : \mathfrak{s} \rightarrow$$ Der$$(\mathfrak{h})$$ is a representation on $$\mathfrak{h}$$-derivations;
3) $$\phi$$ is not of type (I) and $$\phi(\mathfrak{s})$$ is a nilpotent subalgebra of Der$$(\mathfrak{h})$$.\smallbreak The first result of the paper shows the nonexistence of Hermitian-symplectic and SKT structures on quotients of splitting Lie groups: let $$G$$ be endowed with a left-invariant complex structure $$J$$ such that
a) $$G$$ is splitting;
b) $$\mathfrak{h}$$ is $$J$$-invariant;
c) $$J_{|h} \circ \phi(X) = \phi(X) \circ J_{|h}$$ for any $$X \in \mathfrak{s}$$.
Then $$\mathfrak{g}$$ does not admit taming symplectic structures. Moreover, if $$\mathfrak{s}$$ is nilpotent and $$J$$-invariant, then $$\mathfrak{g}$$ does not even have $$J$$-Hermitian SKT metrics.

Instances satisfying the above requirements are abundant, and relevantly include the so-called Oeljeklaus-Toma manifolds [K. Oeljeklaus and M. Toma, Ann. Inst. Fourier 55, No. 1, 161–171 (2005; Zbl 1071.32017)].
But since most simply connected solvable Lie groups are not splitting, the authors cover the complementary situation by proving the following. Let $$(G/\Gamma, J)$$ be a complex solvmanifold and suppose $$J$$ is invariant under the action of a nilpotent complement of the nilradical $$\mathfrak{n}$$. Then $$G/\Gamma$$ is Hermitian-symplectic if and only if $$J$$ is Kähler. The same holds for Abelian complex structures.
Finally, the almost Abelian situation (the nilradical $$\mathfrak{n}$$ is Abelian of codimension 1) is discussed. Non-existence of taming symplectic forms is proved when $$(G/\Gamma, J)$$ is a complex solvmanifold with $$G$$ almost Abelian, $$\mathfrak{g}$$ not of type (I) or six-dimensional.

##### MSC:
 32Q15 Kähler manifolds 53C55 Global differential geometry of Hermitian and Kählerian manifolds 53C30 Differential geometry of homogeneous manifolds
##### Keywords:
special Hermitian metrics; solvmanifolds
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