scholarly journals The complex geometry of two exceptional flag manifolds

2020 ◽  
Vol 199 (6) ◽  
pp. 2227-2241
Author(s):  
D. Kotschick ◽  
D. K. Thung
Keyword(s):  
Lie Group ◽  
Complex Geometry ◽  
Complex Structure ◽  
The Other ◽  
Flag Manifolds ◽  
Dimensional Sphere ◽  
Complex Structures ◽  
Almost Complex Structures ◽  
Almost Complex ◽  
Chern Numbers

Abstract We discuss the complex geometry of two complex five-dimensional Kähler manifolds which are homogeneous under the exceptional Lie group $$G_2$$ G 2 . For one of these manifolds, rigidity of the complex structure among all Kählerian complex structures was proved by Brieskorn; for the other one, we prove it here. We relate the Kähler assumption in Brieskorn’s theorem to the question of existence of a complex structure on the six-dimensional sphere, and we compute the Chern numbers of all $$G_2$$ G 2 -invariant almost complex structures on these manifolds.

scholarly journals Invariant almost complex geometry on flag manifolds: geometric formality and Chern numbers

2016 ◽  
Vol 196 (1) ◽  
pp. 165-200 ◽  
Author(s):  
Lino Grama ◽  
Caio J. C. Negreiros ◽  
Ailton R. Oliveira
Keyword(s):  
Complex Geometry ◽  
Flag Manifolds ◽  
Almost Complex ◽  
Chern Numbers ◽  
Geometric Formality

scholarly journals Families of(1,2)-symplectic metrics on full flag manifolds

2002 ◽  
Vol 29 (11) ◽  
pp. 651-664 ◽  
Author(s):  
Marlio Paredes
Keyword(s):  
Flag Manifolds ◽  
Invariant Metrics ◽  
Complex Structures ◽  
Symplectic Invariant ◽  
Almost Complex Structures ◽  
Almost Complex ◽  
Complex Flag ◽  
Complex Flag Manifolds

We obtain new families of(1,2)-symplectic invariant metrics on the full complex flag manifoldsF(n). Forn≥5, we characterizen−3differentn-dimensional families of(1,2)-symplectic invariant metrics onF(n). Each of these families corresponds to a different class of nonintegrable invariant almost complex structures onF(n).

scholarly journals BRANCHED SHADOWS AND COMPLEX STRUCTURES ON 4-MANIFOLDS

2008 ◽  
Vol 17 (11) ◽  
pp. 1429-1454 ◽  
Author(s):  
FRANCESCO COSTANTINO
Keyword(s):  
Complex Structure ◽  
Complex Structures ◽  
Almost Complex Structure ◽  
Homotopy Classes ◽  
Almost Complex Structures ◽  
Almost Complex

We define and study branched shadows of 4-manifolds as a combination of branched spines of 3-manifolds and of Turaev's shadows. We use these objects to combinatorially represent 4-manifolds equipped with Spinc-structures and homotopy classes of almost complex structures. We then use branched shadows to study complex 4-manifolds and prove that each almost complex structure on a 4-dimensional handlebody is homotopic to a complex one.

On the Relation of Real and Complex Lie Supergroups

2015 ◽  
Vol 58 (2) ◽  
pp. 281-284 ◽  
Author(s):  
Matthias Kalus
Keyword(s):  
Complex Structure ◽  
Sufficient Conditions ◽  
Complex Structures ◽  
Almost Complex Structure ◽  
New Approach ◽  
Almost Complex Structures ◽  
Almost Complex ◽  
Lie Supergroups ◽  
Necessary And Sufficient

AbstractA complex Lie supergroup can be described as a real Lie supergroup with integrable almost complex structure. The necessary and sufficient conditions on an almost complex structure on a real Lie supergroup for defining a complex Lie supergroup are deduced. The classification of real Lie supergroups with such almost complex structures yields a new approach to the known classification of complex Lie supergroups by complexHarish-Chandra superpairs. A universal complexi ûcation of a real Lie supergroup is constructed

scholarly journals On Kodaira dimension of almost complex 4-dimensional solvmanifolds without complex structures

2021 ◽  
pp. 2150075
Author(s):  
Andrea Cattaneo ◽  
Antonella Nannicini ◽  
Adriano Tomassini
Keyword(s):  
Complex Structure ◽  
Kodaira Dimension ◽  
Classification Theory ◽  
Complex Structures ◽  
Almost Complex Structure ◽  
Almost Complex Manifolds ◽  
Kähler Structure ◽  
Almost Complex Structures ◽  
Almost Complex ◽  
Almost Kähler

The aim of this paper is to continue the study of Kodaira dimension for almost complex manifolds, focusing on the case of compact [Formula: see text]-dimensional solvmanifolds without any integrable almost complex structure. According to the classification theory we consider: [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text]. For the first solvmanifold we introduce special families of almost complex structures, compute the corresponding Kodaira dimension and show that it is no longer a deformation invariant. Moreover, we prove Ricci flatness of the canonical connection for the almost Kähler structure. Regarding the other two manifolds we compute the Kodaira dimension for certain almost complex structures. Finally, we construct a natural hypercomplex structure providing a twistorial description.

Harmonicity of proper almost complex structures on Walker 4-manifolds

2017 ◽  
Vol 14 (06) ◽  
pp. 1750094
Author(s):  
Johann Davidov ◽  
Absar Ul-Haq ◽  
Oleg Mushkarov
Keyword(s):  
Twistor Space ◽  
Harmonic Map ◽  
Complex Structure ◽  
Complex Structures ◽  
Almost Complex Structure ◽  
Almost Complex Structures ◽  
Almost Complex ◽  
Harmonic Section

Every Walker [Formula: see text]-manifold [Formula: see text], endowed with a canonical neutral metric [Formula: see text], admits a specific almost complex structure called proper. In this paper, we find the conditions under which a proper almost complex structure is a harmonic section or a harmonic map from [Formula: see text] to its hyperbolic twistor space.

scholarly journals S.-S. Chern’s study of almost-complex structures on the six-sphere

2021 ◽  
pp. 2140006
Author(s):  
Robert L. Bryant
Keyword(s):  
Open Problem ◽  
Complex Structure ◽  
Complex Structures ◽  
Almost Complex Structure ◽  
Exceptional Group ◽  
Almost Complex Structures ◽  
Almost Complex ◽  
Interesting Class

In April 2003, Chern began a study of almost-complex structures on the six-sphere, with the idea of exploiting the special properties of its well-known almost-complex structure invariant under the exceptional group [Formula: see text]. While he did not solve the (currently still open) problem of determining whether there exists an integrable almost-complex structure on [Formula: see text], he did prove a significant identity that resolves the question for an interesting class of almost-complex structures on [Formula: see text].

scholarly journals Symmetries of almost complex structures and pseudoholomorphic foliations

2014 ◽  
Vol 25 (08) ◽  
pp. 1450079 ◽  
Author(s):  
Boris Kruglikov
Keyword(s):  
Complex Structure ◽  
Complex Structures ◽  
Almost Complex Structure ◽  
Integrable Structure ◽  
Infinite Dimensional ◽  
Almost Complex Structures ◽  
Almost Complex ◽  
Finite Dimensionality ◽  
Local Symmetries ◽  
Symmetric Structures

Contrary to complex structures, a generic almost complex structure J has no local symmetries. We give a criterion for finite-dimensionality of the pseudogroup G of local symmetries for a given almost complex structure J. It will be indicated that a large symmetry pseudogroup (infinite-dimensional) is a signature of some integrable structure, like a pseudoholomorphic foliation. We classify the sub-maximal (from the viewpoint of the size of G) symmetric structures J. Almost complex structures in dimensions 4 and 6 are discussed in greater details in this paper.

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