Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T06:44:29.706Z Has data issue: false hasContentIssue false
  • English
  • Français

On generalized Gauduchon metrics

Part of: Compact analytic spaces Global differential geometry

Published online by Cambridge University Press:  21 March 2013

Anna Fino  and
Luis Ugarte
Anna Fino
Affiliation:
Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy ([email protected])
Luis Ugarte
Affiliation:
Departamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, Campus Plaza San Francisco, 50009 Zaragoza, Spain ([email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study a class of Hermitian metrics on complex manifolds, recently introduced by Fu, Wang and Wu, which are a generalization of Gauduchon metrics. This class includes the class of Hermitian metrics for which the associated fundamental 2-form is ∂∂-closed. Examples are given on nilmanifolds, on products of Sasakian manifolds, on S1-bundles and via the twist construction introduced by Swann.

Keywords

Hermitian structureGauduchon metricKähler manifolds

MSC classification

Secondary: 53C55: Hermitian and Kählerian manifolds 32J27: Compact Kähler manifolds 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 3 , October 2013 , pp. 733 - 753
Copyright
Copyright © Edinburgh Mathematical Society 2013 

References

1.Alexandrov, B. and Ivanov, S., Vanishing theorems on Hermitian manifolds, Diff. Geom. Applic. 14 (2001), 251–265.CrossRefGoogle Scholar
2.Andrada, A., Fino, A. and Vezzoni, L., A class of Sasakian 5-manifolds, Transform. Groups 14 (2009), 493–512.CrossRefGoogle Scholar
3.Apostolov, V. and Gualtieri, M., Generalized Kähler manifolds with split tangent bundle, Commun. Math. Phys. 271 (2007), 561–575.CrossRefGoogle Scholar
4.Blanchard, A., Sur les variétés analytiques complexes, Annales Scient. Éc. Norm. Sup. 73 (1956), 157–202.CrossRefGoogle Scholar
5.Enrietti, N., Fino, A. and Vezzoni, L., Tamed symplectic forms and SKT metrics, J. Symplect. Geom. 10 (2012), 203–223.CrossRefGoogle Scholar
6.Fernández, M., Fino, A., Ugarte, L. and Villacampa, R., Strong Kähler with torsion structures from almost contact manifolds, Pac. J. Math. 249 (2011), 49–75.CrossRefGoogle Scholar
7.Fino, A. and Grantcharov, G., Properties of manifolds with skew-symmetric torsion and special holonomy, Adv. Math. 189 (2004), 439–450.CrossRefGoogle Scholar
8.Fino, A. and Tomassini, A., Blow-ups and resolutions of strong Kähler with torsion metrics, Adv. Math. 221 (2009), 914–935.CrossRefGoogle Scholar
9.Fino, A. and Tomassini, A., On astheno-Kähler metrics, J. Lond. Math. Soc. 83 (2011), 290–308.CrossRefGoogle Scholar
10.Fino, A., Parton, M. and Salamon, S., Families of strong KT structures in six dimensions, Comment. Math. Helv. 79 (2004), 317–340.CrossRefGoogle Scholar
11.Fu, J., Wang, Z. and Wu, D., Semilinear equations, the γκ function, and generalized Gauduchon metrics, J. Eur. Math. Soc., in press.Google Scholar
12.Gates, S. J., Hull, C. M. and Roček, M., Twisted multiplets and new supersymmetric nonlinear sigma models, Nucl. Phys. B 248 (1984), 157–186.CrossRefGoogle Scholar
13.Gauduchon, P., La 1-forme de torsion d'une variété hermitienne compacte, Math. Annalen 267 (1984), 495–518.CrossRefGoogle Scholar
14.Grantcharov, D., Grantcharov, G. and Poon, Y. S., Calabi–Yau connections with torsion on toric bundles, J. Diff. Geom. 78 (2008), 13–32.Google Scholar
15.Gray, A. and Hervella, L. M., The sixteen classes of almost Hermitian manifolds and their linear invariants, Annali Mat. Pura Appl. 123 (1980), 35–58.CrossRefGoogle Scholar
16.Gualtieri, M., Generalized complex geometry, DPhil thesis, University of Oxford (2003; e-print, math.DG/0401221).Google Scholar
17.Hitchin, N. J., Instantons and generalized Kähler geometry, Commun. Math. Phys. 265 (2006), 131–164.CrossRefGoogle Scholar
18.Ivanov, S. and Papadopoulos, G., Vanishing theorems and string backgrounds, Class. Quant. Grav. 18 (2001), 1089–1110.CrossRefGoogle Scholar
19.Jost, J. and Yau, S.-T., A non-linear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry, Acta Math. 170 (1993), 221–254 (corrigendum: Acta Math. 173 (1994), 307).CrossRefGoogle Scholar
20.Kobayashi, S., Principal fibre bundles with the 1-dimensional toroidal group, Tohoku Math. J. 8 (1956), 29–45.CrossRefGoogle Scholar
21.Matsuo, K., Astheno–Kähler structures on Calabi–Eckmann manifolds, Colloq. Math. 115 (2009), 33–39.CrossRefGoogle Scholar
22.Matsuo, K. and Takahashi, T., On compact astheno-Kähler manifolds, Colloq. Math. 89 (2001), 213–221.CrossRefGoogle Scholar
23.Ogawa, Y., Some properties on manifolds with almost contact structures, Tohoku Math. J. 15 (1963), 148–161.CrossRefGoogle Scholar
24.Papadopolous, G., KT and HKT geometries in strings and in black hole moduli spaces, in Proc. Workshop on Special Geometric Structures in String Theory, Bonn, 8–11 September 2011 (ed. Alekseevsky, S. V., Cortés, V., Devchand, Ch. and Van Proeyen, A.; available at www.emis.de/proceedings/SGSST2001/proc2001.html).Google Scholar
25.Salamon, S., Complex structures on nilpotent Lie algebras, J. Pure Appl. Alg. 157 (2001), 311–333.CrossRefGoogle Scholar
26.Streets, J. and Tian, G., A parabolic flow of pluriclosed metrics, Int. Math. Res. Not. 16 (2010), 3101–3133.Google Scholar
27.Strominger, A., Superstrings with torsion, Nuclear Phys. B 274 (1986), 253–284.CrossRefGoogle Scholar
28.Swann, A., Twisting Hermitian and hypercomplex geometries, Duke Math. J. 155 (2010), 403–431.CrossRefGoogle Scholar
29.Tsukada, K., Eigenvalues of the Laplacian on Calabi–Eckmann manifolds, J. Math. Soc. Jpn 33 (1981), 673–691.CrossRefGoogle Scholar
30.Ugarte, L., Hermitian structures on six dimensional nilmanifolds, Transform. Groups 12 (2007), 175–202.CrossRefGoogle Scholar
31.Ugarte, L. and Villacampa, R., Non-nilpotent complex geometry of nilmanifolds and heterotic supersymmetry, eprint (arXiv:0912.5110v1 [math.DG]).Google Scholar