On the existence of balanced and SKT metrics on nilmanifolds
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- by Anna Fino and Luigi Vezzoni
- Proc. Amer. Math. Soc. 144 (2016), 2455-2459
- DOI: https://doi.org/10.1090/proc/12954
- Published electronically: January 27, 2016
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Abstract:
On a complex manifold a Hermitian metric which is simultaneously SKT and balanced has to be necessarily Kähler. It has been conjectured that if a compact complex manifold $(M, J)$ has an SKT metric and a balanced metric both compatible with $J$, then $(M, J)$ is necessarily Kähler. We show that the conjecture is true for nilmanifolds.References
- Lucia Alessandrini and Giovanni Bassanelli, Metric properties of manifolds bimeromorphic to compact Kähler spaces, J. Differential Geom. 37 (1993), no. 1, 95–121. MR 1198601
- Lucia Alessandrini and Giovanni Bassanelli, Modifications of compact balanced manifolds, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 12, 1517–1522 (English, with English and French summaries). MR 1340064
- A. Andrada and R. Villacampa, Abelian balanced hermitian structures on unimodular Lie algebras, to apper in Transform. Groups, arXiv: 1412.7092.
- Bogdan Alexandrov and Stefan Ivanov, Vanishing theorems on Hermitian manifolds, Differential Geom. Appl. 14 (2001), no. 3, 251–265. MR 1836272, DOI 10.1016/S0926-2245(01)00044-4
- Chal Benson and Carolyn S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), no. 4, 513–518. MR 976592, DOI 10.1016/0040-9383(88)90029-8
- Jean-Michel Bismut, A local index theorem for non-Kähler manifolds, Math. Ann. 284 (1989), no. 4, 681–699. MR 1006380, DOI 10.1007/BF01443359
- Ionuţ Chiose, Obstructions to the existence of Kähler structures on compact complex manifolds, Proc. Amer. Math. Soc. 142 (2014), no. 10, 3561–3568. MR 3238431, DOI 10.1090/S0002-9939-2014-12128-9
- Nicola Enrietti, Anna Fino, and Luigi Vezzoni, Tamed symplectic forms and strong Kähler with torsion metrics, J. Symplectic Geom. 10 (2012), no. 2, 203–223. MR 2926995
- Anna Fino and Gueo Grantcharov, Properties of manifolds with skew-symmetric torsion and special holonomy, Adv. Math. 189 (2004), no. 2, 439–450. MR 2101226, DOI 10.1016/j.aim.2003.10.009
- Anna Fino, Maurizio Parton, and Simon Salamon, Families of strong KT structures in six dimensions, Comment. Math. Helv. 79 (2004), no. 2, 317–340. MR 2059435, DOI 10.1007/s00014-004-0803-3
- Anna Fino and Adriano Tomassini, A survey on strong KT structures, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 52(100) (2009), no. 2, 99–116. MR 2521810
- Keizo Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, 65–71. MR 946638, DOI 10.1090/S0002-9939-1989-0946638-X
- Anna Fino and Luigi Vezzoni, Special Hermitian metrics on compact solvmanifolds, J. Geom. Phys. 91 (2015), 40–53. MR 3327047, DOI 10.1016/j.geomphys.2014.12.010
- D. Grantcharov, G. Grantcharov, and Y. S. Poon, Calabi-Yau connections with torsion on toric bundles, J. Differential Geom. 78 (2008), no. 1, 13–32. MR 2406264
- S. Ivanov and G. Papadopoulos, Vanishing theorems on $(\ell |k)$-strong Kähler manifolds with torsion, Adv. Math. 237 (2013), 147–164. MR 3028575, DOI 10.1016/j.aim.2012.12.019
- Hisashi Kasuya, Minimal models, formality, and hard Lefschetz properties of solvmanifolds with local systems, J. Differential Geom. 93 (2013), no. 2, 269–297. MR 3024307
- M. L. Michelsohn, On the existence of special metrics in complex geometry, Acta Math. 149 (1982), no. 3-4, 261–295. MR 688351, DOI 10.1007/BF02392356
- Jeffrey Streets and Gang Tian, A parabolic flow of pluriclosed metrics, Int. Math. Res. Not. IMRN 16 (2010), 3101–3133. MR 2673720, DOI 10.1093/imrn/rnp237
- Jeffrey Streets and Gang Tian, Regularity results for pluriclosed flow, Geom. Topol. 17 (2013), no. 4, 2389–2429. MR 3110582, DOI 10.2140/gt.2013.17.2389
- Jixiang Fu, Jun Li, and Shing-Tung Yau, Balanced metrics on non-Kähler Calabi-Yau threefolds, J. Differential Geom. 90 (2012), no. 1, 81–129. MR 2891478
- Luis Ugarte, Hermitian structures on six-dimensional nilmanifolds, Transform. Groups 12 (2007), no. 1, 175–202. MR 2308035, DOI 10.1007/s00031-005-1134-1
- Misha Verbitsky, Rational curves and special metrics on twistor spaces, Geom. Topol. 18 (2014), no. 2, 897–909. MR 3190604, DOI 10.2140/gt.2014.18.897
Bibliographic Information
- Anna Fino
- Affiliation: Dipartimento di Matematica G. Peano, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
- MR Author ID: 363840
- ORCID: 0000-0003-0048-2970
- Email: annamaria.fino@unito.it
- Luigi Vezzoni
- Affiliation: Dipartimento di Matematica G. Peano, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
- Email: luigi.vezzoni@unito.it
- Received by editor(s): July 2, 2105
- Published electronically: January 27, 2016
- Additional Notes: This work was partially supported by the project PRIN, Varietà reali e complesse: geometria, topologia e analisi armonica, the project FIRB, Differential Geometry and Geometric functions theory and by GNSAGA (Indam) of Italy
- Communicated by: Lei Ni
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 2455-2459
- MSC (2010): Primary 32J27; Secondary 53C55, 53C30, 53D05
- DOI: https://doi.org/10.1090/proc/12954
- MathSciNet review: 3477061