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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.

32Q15 Kähler manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C30 Differential geometry of homogeneous manifolds
Full Text: DOI Euclid arXiv
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