Properties of manifolds with skew-symmetric torsion and special holonomy

Fino, Anna; Grantcharov, Gueo

doi:10.1016/j.aim.2003.10.009

Cited by 114 publications

(142 citation statements)

References 24 publications

Supporting

Mentioning

140

Contrasting

Order By: Relevance

“…Thus, a closed (equivalently, holomorphic) (n, 0)-form of constant norm is necessarily parallel with respect to the Chern connection. For more details, see [2,9,20].…”

Section: Cmhmentioning

confidence: 99%

Families of strong KT structures in six dimensions

Fino¹,

Parton²,

Salamon³

2004

Commentarii Mathematici Helvetici

Self Cite

132

193

View full text Add to dashboard Cite

Abstract. This paper classifies Hermitian structures on 6-dimensional nilmanifolds M = Γ\G for which the fundamental 2-form is ∂∂-closed, a condition that is shown to depend only on the underlying complex structure J of M . The space of such J is described when G is the complex Heisenberg group, and explicit solutions are obtained from a limaçon-shaped curve in the complex plane. Related theory is used to provide examples of various types of Ricci-flat structures. Mathematics Subject Classification (2000). 53C55; 32G05, 17B30, 81T30.

show abstract

“…Thus, a closed (equivalently, holomorphic) (n, 0)-form of constant norm is necessarily parallel with respect to the Chern connection. For more details, see [2,9,20].…”

Section: Cmhmentioning

confidence: 99%

Families of strong KT structures in six dimensions

Fino¹,

Parton²,

Salamon³

2004

Commentarii Mathematici Helvetici

Self Cite

132

193

View full text Add to dashboard Cite

show abstract

“…This condition has been much studied in the mathematical physics literature [2,21,24,38,42,45,61], and Conjecture 4.1 would provide many more examples of such special metrics.…”

Section: Canonical Metrics On Non-kähler Calabi-yau Manifoldsmentioning

confidence: 99%

“…A lot of interest in the subject was generated by "Reid's fantasy" [64] that all Calabi-Yau threefolds with trivial canonical bundle should form a connected family provided one allows deformations and singular transitions through non-Kähler manifolds with trivial canonical bundle. The geometry of compact complex manifolds with trivial canonical bundle has been investigated for example by [2,6,12,19,21,25,26,27,30,33,45,52,62,65] and others. In this paper we will consider a more general class of manifolds, that we now define, and argue that they can naturally be considered as "non-Kähler Calabi-Yau" manifolds.…”

mentioning

confidence: 99%

Non-Kähler Calabi-Yau manifolds

Tosatti¹

2015

Analysis, Complex Geometry, and Mathematical Physics

View full text Add to dashboard Cite

Abstract. We study the class of compact complex manifolds whose first Chern class vanishes in the Bott-Chern cohomology. This class includes all manifolds with torsion canonical bundle, but it is strictly larger. After making some elementary remarks, we show that a manifold in Fujiki's class C with vanishing first Bott-Chern class has torsion canonical bundle. We also give some examples of non-Kähler CalabiYau manifolds, and discuss the problem of defining and constructing canonical metrics on them. IntroductionIn this paper, Calabi-Yau manifolds are defined to be compact Kähler manifolds M with c 1 (M ) = 0 in H 2 (M, R). Thanks to Yau's theorem [84] these are precisely the compact manifolds that admit Ricci-flat Kähler metrics. Using this, Calabi proved a decomposition theorem [13] which shows that any such manifold has a finite unramified cover which splits as a product of a torus and a Calabi-Yau manifold with vanishing first Betti number. From this one can easily deduce that Calabi-Yau manifolds have holomorphically torsion canonical bundle (see Theorem 1.4).One can ask how much of this theory carries over to the case of non-Kähler Hermitian manifolds. Simple examples, such as a Hopf surface diffeomorphic to S 1 × S 3 , show that the condition that c 1 (M ) = 0 in H 2 (M, R) is too weak in general (see Example 3.3). On the other hand, much interest has been devoted to studying non-Kähler compact complex manifolds with holomorphically trivial (or more generally torsion) canonical bundle, and many examples can be found in [6,15,17,20,23,33,35,36,37,40,43,53,60,78,83,86] and references therein. For example, every compact complex nilmanifold with a left-invariant complex structure has trivial canonical bundle, and it is always non-Kähler unless it is a torus [6]. A lot of interest in the subject was generated by "Reid's fantasy" [64] that all Calabi-Yau threefolds with trivial canonical bundle should form a connected family provided one allows deformations and singular transitions through non-Kähler manifolds with trivial canonical bundle. The geometry of compact complex manifolds with trivial canonical bundle has been investigated for example by [2,6,12,19,21,25,26,27,30,33,45,52,62,65] and others. In this paper we will consider a more general class of manifolds, that we now define, and argue that they can naturally be considered as "non-Kähler Calabi-Yau" manifolds.On any compact complex manifold there is a (finite-dimensional) cohomology theory called Bott-Chern cohomology. We will need only the real (1, 1) Bott-Chern cohomologyThere is a "first Bott-Chern class" map c BC 1 : Pic(M ) → H 1,1 BC (M, R), which can be described as follows. Given any holomorphic line bundle L → M and any Hermitian metric h on the fibers of L, its curvature form R h is locally given by − √ −1∂∂ log h. Then R h is a closed real (1, 1)-form and if we choose a different metricIf g is any Hermitian metric on M , with fundamental 2-form ω, then its first Chern form given locally byWe will call Ric(ω) the Chern-Ricci form of ω.We the...

show abstract

“…In fact, there exist hypercomplex manifolds of dimension ≥ 8 which do not admit any HKT metric compatible with the hypercomplex structure [13,6]. These manifolds are nilmanifolds, i.e.…”

Section: Introductionmentioning

confidence: 99%

New HKT manifolds arising from quaternionic representations

Barberis

Fino

2009

Math. Z.

View full text Add to dashboard Cite

Abstract. We give a procedure for constructing an 8n-dimensional HKT Lie algebra starting from a 4n-dimensional one by using a quaternionic representation of the latter. The strong (respectively, weak, hyper-Kähler, balanced) condition is preserved by our construction. As an application of our results we obtain a new compact HKT manifold with holonomy in SL(n, H) which is not a nilmanifold. We find in addition new compact strong HKT manifolds.We also show that every Kähler Lie algebra equipped with a flat, torsion-free complex connection gives rise to an HKT Lie algebra. We apply this method to two distinguished 4-dimensional Kähler Lie algebras, thereby obtaining two conformally balanced HKT metrics in dimension 8.Both techniques prove to be an effective tool for giving the explicit expression of the corresponding HKT metrics.

show abstract

Properties of manifolds with skew-symmetric torsion and special holonomy

Cited by 114 publications

References 24 publications

Families of strong KT structures in six dimensions

Families of strong KT structures in six dimensions

Non-Kähler Calabi-Yau manifolds

New HKT manifolds arising from quaternionic representations

Contact Info

Product

Resources

About