On the nonexistence of S6 type complex threefolds in any compact homogeneous complex manifolds with the compact lie group G2 as the base manifold

2020 ◽  
Vol 305 (2) ◽  
pp. 641-644
Author(s):  
Daniel Guan
Keyword(s):  
Lie Group ◽  
Complex Manifolds ◽  
Compact Lie Group ◽  
Base Manifold ◽  
Type Complex ◽  
Homogeneous Complex

scholarly journals On closed radical orbits in homogeneous complex manifolds

1996 ◽  
Vol 54 (3) ◽  
pp. 363-368 ◽  
Author(s):  
Bruce Gilligan
Keyword(s):  
Finite Number ◽  
Lie Group ◽  
Discrete Subgroup ◽  
Connected Components ◽  
Complex Manifolds ◽  
Algebraic Subgroup ◽  
Homogeneous Complex ◽  
Semisimple Complex ◽  
Klein Form

Suppose G is a complex Lie group having a finite number of connected components and H is a closed complex subgroup of G with H° solvable. Let RG denote the radical of G. We show the existence of closed complex subgroups I and J of G containing H such that I/H is a connected solvmanifold with I° ⊃ RG, the space G/J has a Klein form SG/A, where A is an algebraic subgroup of the semisimple complex Lie group SG: = G/RG, and, unless I = J, the space J/I has Klein form , where is a Zariski dense discrete subgroup of some connected positive dimensional semisimple complex Lie group Ŝ.

Homogeneous Complex Manifolds with more than One End

1989 ◽  
Vol 41 (1) ◽  
pp. 163-177 ◽  
Author(s):  
B. Gilligan ◽  
K. Oeljeklaus ◽  
W. Richthofer
Keyword(s):  
Lie Group ◽  
Homogeneous Spaces ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Complex Manifolds ◽  
Connected Subgroup ◽  
Homogeneous Complex ◽  
Homogeneous Complex Manifold ◽  
Homogeneous Cone ◽  
Closed Connected Subgroup

For homogeneous spaces of a (real) Lie group one of the fundamental results concerning ends (in the sense of Freudenthal [8] ) is due to A. Borel [6]. He showed that if X = G/H is the homogeneous space of a connected Lie group G by a closed connected subgroup H, then X has at most two ends. And if X does have two ends, then it is diffeomorphic to the product of R with the orbit of a maximal compact subgroup of G.In the setting of homogeneous complex manifolds the basic idea should be to find conditions which imply that the space has at most two ends and then, when the space has exactly two ends, to display the ends via bundles involving C* and compact homogeneous complex manifolds. An analytic condition which ensures that a homogeneous complex manifold X has at most two ends is that X have non-constant holomorphic functions and the structure of such a space with exactly two ends is determined, namely, it fibers over an affine homogeneous cone with its vertex removed with the fiber being compact [9], [13].

scholarly journals RATIONAL LOCAL SYSTEMS AND CONNECTED FINITE LOOP SPACES

2021 ◽  
pp. 1-29
Author(s):  
DREW HEARD
Keyword(s):  
Lie Group ◽  
Loop Space ◽  
Algebraic Model ◽  
Compact Lie Group ◽  
Loop Spaces ◽  
Local Systems ◽  
Special Case ◽  
Finite Loop ◽  
Finite Loop Space

Abstract Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree G-spectra. More generally, we show that if K is a closed subgroup of a compact Lie group G such that the Weyl group W G K is connected, then a certain category of rational G-spectra “at K” has an algebraic model. For example, when K is the trivial group, this is just the category of rational cofree G-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

scholarly journals Stratifications of equivariant varieties

1977 ◽  
Vol 16 (2) ◽  
pp. 279-295 ◽  
Author(s):  
M.J. Field
Keyword(s):  
Lie Group ◽  
General Position ◽  
Higher Order ◽  
Order Conditions ◽  
Compact Lie Group ◽  
Algebraic Varieties ◽  
Equivariant Maps

Let G be a compact Lie group and V and W be linear G spaces. A study is made of the canonical stratification of some algebraic varieties that arise naturally in the theory of C∞ equivariant maps from V to W. The main corollary of our results is the equivalence of Bierstone's concept of “equivariant general position” with our own of “G transversal”. The paper concludes with a description of Bierstone's higher order conditions for equivariant maps in the framework of equisingularity sequences.

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