×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] A. Andrada, M. L. Barberis and I. Dotti, Classification of abelian complex structures on 6-dimensional Lie algebras, J. Lond. Math. Soc. (2) 83 (2011), no. 1, 232-255. · Zbl 1218.17006 · doi:10.1112/jlms/jdq071 · arxiv:0908.3213
[2] A. Andrada, M. L. Barberis and I. Dotti, Abelian Hermitian Geometry, Differential Geom. Appl. 30 (2012), 509-519. · Zbl 1253.53025 · doi:10.1016/j.difgeo.2012.07.001
[3] L. Auslander, An exposition of the structure of solvmanifolds. I, Algebraic theory, Bull. Amer. Math. Soc. 79 (1973), no. 2, 227-261. · Zbl 0265.22016 · doi:10.1090/S0002-9904-1973-13134-9
[4] M. L. Barberis, I. G. Dotti and R. J. Miatello, On certain locally homogeneous Clifford manifolds, Ann. Glob. Anal. Geom. 13 (1995), 513-518. · Zbl 0832.53039 · doi:10.1007/BF00773661
[5] O. Baues and J. Riesterer, Virtually abelian Kähler and projective groups, Abh. Math. Semin. Univ. Hambg. 81 (2011), no. 2, 191-213. · Zbl 1254.32029 · doi:10.1007/s12188-011-0056-1 · arxiv:0911.2459
[6] D. Burde and W. A. de Graaf, Classification of Novikov algebras, Appl. Algebra Engrg. Comm. Comput. 24 (2013), no. 1, 1-15. · Zbl 1301.17023 · doi:10.1007/s00200-012-0180-x · arxiv:1106.5954
[7] G. Cavalcanti, SKT geometry, arXiv: · arxiv.org
[8] B. Y. Chu, Symplectic homogeneous spaces, Trans. Amer. Math. Soc. 197 (1974), 145-159. · Zbl 0261.53039 · doi:10.2307/1996932
[9] S. Console, A. Fino and H. Kasuya, Modification and cohomology of solvmanifolds, arXiv: · Zbl 1352.22007 · arxiv.org
[10] K. Dekimpe, Solvable Lie algebras, Lie groups and polynomial structures, Compositio Math. 121 (2000), no. 2, 183-204. · Zbl 1002.22003 · doi:10.1023/A:1001738932743
[11] K. Dekimpe, Semi-simple splittings for solvable Lie groups and polynomial structures, Forum Math. 12 (2000), no. 1, 77-96. · Zbl 0946.22009 · doi:10.1515/form.1999.030
[12] J. Dixmier, Cohomologie des algebres de Lie nilpotentes, Acta Sci. Math. Szeged 16 (1955), 246-250. · Zbl 0066.02403
[13] T. Draghici, T.-J. Li and W. Zhang, On the \(J\)-anti-invariant cohomology of almost complex 4-manifolds, Q. J. Math. 64 (2013), no. 1, 83-111. · Zbl 1271.32029 · doi:10.1093/qmath/har034 · arxiv:1104.2511
[14] N. Enrietti and A. Fino, Special Hermitian metrics and Lie groups, Differential Geom. Appl. 29 (2011), suppl. 1, 211-219. · Zbl 1247.32024 · doi:10.1016/j.difgeo.2011.04.043 · arxiv:1104.1612
[15] N. Enrietti, A. Fino and L. Vezzoni, Tamed symplectic forms and strong Kähler with torsion metrics, J. Symplectic Geom. 10 (2012), no. 2, 203-223 · Zbl 1248.53070 · doi:10.4310/JSG.2012.v10.n2.a3 · euclid:jsg/1339096435 · arxiv:1002.3099
[16] A. Fino, M. Parton and S. Salamon, Families of strong KT structures in six dimensions, Comment. Math. Helv. 79 (2004), no. 2, 317-340. · Zbl 1062.53062 · doi:10.1007/s00014-004-0803-3 · arxiv:math/0209259
[17] A. Fino and A. Tomassini, A survey on strong KT structures, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 52 (100) (2009), no. 2, 99-116. · Zbl 1199.53138
[18] K. Hasegawa, A note on compact solvmanifolds with Kähler structures, Osaka J. Math. 43 (2006), no. 1, 131-135. · Zbl 1105.32017 · arxiv:math/0406227
[19] H. Kasuya, Minimal models, formality and hard Lefschetz properties of solvmanifolds with local systems, J. Differential Geom. 93 (2013), no 2, 269-298. · Zbl 1373.53069
[20] H. Kasuya, Vaisman metrics on solvmanifolds and Oeljeklaus-Toma manifolds, Bull. London Math. Soc. 45 (2013), 15-26. · Zbl 1262.53061 · doi:10.1112/blms/bds057 · arxiv:1204.1878
[21] H. Kasuya, Hodge symmetry and decomposition on non-Kähler solvmanifolds, J. Geom. Phys. 76 (2014), 61-65. · Zbl 1279.22010 · doi:10.1016/j.geomphys.2013.10.012 · arxiv:1109.5929
[22] T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom. 17 (2009), no. 4, 651-683. · Zbl 1225.53066 · doi:10.4310/CAG.2009.v17.n4.a4 · arxiv:0708.2520
[23] K. Oeljeklaus and M. Toma, Non-Kähler compact complex manifolds associated to number fields, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 1, 161-171. · Zbl 1071.32017 · doi:10.5802/aif.2093 · numdam:AIF_2005__55_1_161_0 · eudml:116182
[24] G. Ovando, Invariant pseudo-Kähler metrics in dimension four, J. Lie Theory 16 (2006), 371-391. · Zbl 1102.32011
[25] A. L. Onishchik and E. B. Vinberg (Eds), Lie groups and Lie algebras II, Encyclopaedia of Mathematical Sciences, 41. Springer-Verlag, Berlin, 1994.
[26] T. Peternell, Algebraicity criteria for compact complex manifolds, Math. Ann. 275 (1986), no. 4, 653-672. · Zbl 0606.32018 · doi:10.1007/BF01459143 · eudml:164176
[27] J. E. Snow, Invariant complex structures on four dimensional solvable real Lie groups, Manuscripta Math. 66 (1990), 397-412. · Zbl 0715.22008 · doi:10.1007/BF02568505 · eudml:155481
[28] J. Streets and G. Tian, A parabolic flow of pluriclosed metrics, Int. Math. Res. Not. IMRN 2010 (2010), 3101-3133. · Zbl 1198.53077 · doi:10.1093/imrn/rnp237 · arxiv:0903.4418
[29] M. Verbitsky, Rational curves and special metrics on twistor spaces, arXiv: · Zbl 1300.53053 · doi:10.2140/gt.2014.18.897 · arxiv:1210.6725
[30] T. Yamada, Ricci flatness of certain compact pseudo-Kähler solvmanifolds, J. Geom. Phys. 62 (2012), no. 5, 1338-1345. · Zbl 1239.53100 · doi:10.1016/j.geomphys.2011.06.006
[31] W. Zhang, From Taubes currents to almost Kähler forms, Math. Ann. 356 (2013), no. 9, 969-978. · Zbl 1280.53072 · doi:10.1007/s00208-012-0878-x
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.