Families of strong KT structures in six dimensions

By Anna Fino, Maurizio Parton, S. Salamon. 2002

191 citations recorded by the source corpus.

Abstract This paper classifies Hermitian structures on 6-dimensional nilmanifolds $M=\Ga\bs G$ for which the fundamental 2-form is $\pd\opd$-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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/272f85280cc7e405e210893507a25a9af70b5421

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Blow-ups and resolutions of strong K\"ahler with torsion metrics (2008, background): “7, the blow-up of the strong KT nilmanifolds given in [15] cannot admit any Kähler structure.”
New aspects of the ddc-lemma (2005, unknown): “Using T-duality and results of Fino et al on SKT structures [32], we prove that no 6-nilmanifold admits invariant generalized Kähler structures.”
Geometric Model for Complex Non-Kähler Manifolds with SU (3) Structure (2002, background): “This means that the holonomy of the connection ∇ is contained in SU(n) and this implies that the generalized Ricci form vanishes (see [28], p.”
The long‐time behavior of the homogeneous pluriclosed flow (2017, background): “In dimension 6, all nilpotent examples were classified in [9] (see also [34]), and some partial results on the solvable case are given in [8].”
A survey on strong KT structures (2009, methodology): “Other examples, besides the compact semisimple Lie groups, of homogeneous strong KT manifolds have been found in [17] and they are 6-dimensional nilmanifolds Γ\G, i.”
Ricci Curvatures on Hermitian manifolds (2014, background): “See also [1, 3, 4, 11, 12, 23, 33, 34, 42, 58, 59] for some curvature relations on complex surfaces.”
Non-Kähler string backgrounds and their five torsion classes (2002, background): “It is interesting to point out that for any of the 18 classes one can choose a complex structure which is compatible with the metric and which has W4 = 0 [41].”
A correction to “Tamed symplectic forms and strong Kähler with torsion metrics” (2012, methodology): “If dim g = 6, then the claim follows from [13].”
New aspects of the dd c -lemma (2004, methodology): “Using T-duality and results of Fino et al on SKT structures [32], we prove that no 6-nilmanifold admits invariant generalized Kähler structures.”
Special Hermitian metrics on compact solvmanifolds (2015, methodology): “ anti-self-dual Riemannian manifolds [49]. Examples of SKT manifolds are provided by 6-dimensional nilmanifolds in view of the following 4 ANNA FINO AND LUIGI VEZZONI Theorem 3.3 (Fino-Parton-Salamon [18]). Let M = Γ\G be a 6-dimensional nilmanifold endowed with an invariant complex structure J. The”
Balanced Hermitian structures on almost abelian Lie algebras (2020, background): “Another possible generalization is given by strong Kähler with torsion (SKT, also known as pluriclosed) metrics, satisfying ∂∂ω = 0 (see for instance [27, 24]).”
Hermitian structures on six-dimensional nilmanifolds (2004, background): “We show that, up to equivalence of the complex structure, the strong Kähler with torsion structures (J, g) on M are parametrized by the points in a subset of the Euclidean space, in particular the region inside a certain ovaloid corresponds to such structures on the Iwasawa manifold and the region…”
Invariant Complex Structures on 6-Nilmanifolds: Classification, Frölicher Spectral Sequence and Special Hermitian Metrics (2011, background): “Several aspects of this complex geometry have been investigated, as for instance the Dolbeault cohomology [7, 13, 25], complex deformations [6, 8, 20, 26] or the existence of special Hermitian metrics [16, 29].”