scholarly journals ON THE EINSTEIN–KÄHLER METRIC AND THE HOLONOMY OF A LINE BUNDLE

2002 ◽  
Vol 45 (1) ◽  
pp. 83-90
Author(s):  
Kenji Tsuboi
Keyword(s):  
Line Bundle ◽  
Complex Manifold ◽  
Lie Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Base Space ◽  
Subject Classification ◽  
Compact Complex Manifold ◽  
Identity Component ◽  
Determinant Line Bundle

AbstractIn this paper we give a relation between the Futaki invariant for a compact complex manifold $M$ and the holonomy of a determinant line bundle over a loop in the base space of any principal $G$-bundle, where $G$ is the identity component of the maximal compact subgroup of the complex Lie group consisting of all biholomorphic automorphisms of $M$. Using the property of the Futaki invariant, we show that the holonomy is an obstruction to the existence of the Einstein–Kähler metrics on $M$. Our main result is Theorem 2.1.AMS 2000 Mathematics subject classification: Primary 32Q20. Secondary 58J28; 58J52

scholarly journals Stable Higgs bundles over positive principal elliptic fibrations

2018 ◽  
Vol 5 (1) ◽  
pp. 195-201
Author(s):  
Indranil Biswas ◽  
Mahan Mj ◽  
Misha Verbitsky
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Complex Manifold ◽  
Higgs Bundle ◽  
Compact Complex Manifold ◽  
Holomorphic Line Bundle ◽  
Higgs Bundles ◽  
Stable Vector Bundle ◽  
Hermitian Metric ◽  
The Third

AbstractLet M be a compact complex manifold of dimension at least three and Π : M → X a positive principal elliptic fibration, where X is a compact Kähler orbifold. Fix a preferred Hermitian metric on M. In [14], the third author proved that every stable vector bundle on M is of the form L⊕ Π ⃰ B0, where B0 is a stable vector bundle on X, and L is a holomorphic line bundle on M. Here we prove that every stable Higgs bundle on M is of the form (L ⊕ Π ⃰B0, Π ⃰ ɸX), where (B0, ɸX) is a stable Higgs bundle on X and L is a holomorphic line bundle on M.

EXAMPLES OF A NON-VANISHING CONJECTURE OF KAWAMATA

2007 ◽  
Vol 18 (05) ◽  
pp. 527-533
Author(s):  
YU-LIN CHANG
Keyword(s):  
Line Bundle ◽  
Complex Manifold ◽  
Symmetric Spaces ◽  
Sufficient Conditions ◽  
Compact Complex Manifold ◽  
Holomorphic Line Bundle ◽  
Compact Type ◽  
Hermitian Symmetric Spaces ◽  
Canonical Line Bundle ◽  
Positive Holomorphic Line Bundle

Let M be a compact complex manifold with a positive holomorphic line bundle L, and K be its canonical line bundle. We give some sufficient conditions for the non-vanishing of H0(M, K + L). We also show that the criterion can be applied to interesting classes of examples including all compact locally hermitian symmetric spaces of non-compact type, Mostow–Siu [10] surfaces, Kähler threefolds given by Deraux [3] and examples of Zheng [17].

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].

Minimal actions of semisimple groups

1996 ◽  
Vol 16 (4) ◽  
pp. 821-831 ◽  
Author(s):  
Garrett Stuck
Keyword(s):  
Parabolic Subgroup ◽  
Lie Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Semisimple Lie Group ◽  
Semisimple Groups ◽  
Open Problems ◽  
Minimal Action ◽  
Orbit Structure ◽  
Low Dimension

AbstractWe show that a C0 minimal action of a semisimple Lie group without compact factors is either locally free or induced from a minimal action of a proper parabolic subgroup. We describe the orbit structure of the action restricted to a maximal compact subgroup, and then apply this to minimal actions in low dimension. We give some examples, applications, and open problems.

Complex submanifolds, connections and asymptotics

2010 ◽  
Vol 53 (2) ◽  
pp. 373-383
Author(s):  
Tatyana Foth
Keyword(s):  
Line Bundle ◽  
Complex Manifold ◽  
Compact Complex Manifold ◽  
Line Bundles ◽  
Natural Connection ◽  
Complex Submanifolds ◽  
Positive Line Bundle ◽  
Positive Line

AbstractLet L → X be a positive line bundle on a compact complex manifold X. For compact submanifolds Y, S of X and a holomorphic submersion Y → S with compact fibre, we study curvature of a natural connection on certain line bundles on S.

scholarly journals Langlands parameters of derived functor modules and Vogan diagrams

2003 ◽  
Vol 92 (1) ◽  
pp. 31 ◽  
Author(s):  
Paul D. Friedman
Keyword(s):  
Lie Group ◽  
Simple Root ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Classical Groups ◽  
Parabolic Subalgebra ◽  
Finite Center ◽  
Derived Functor ◽  
Langlands Parameters ◽  
Reductive Lie Group

Let $G$ be a linear reductive Lie group with finite center, let $K$ be a maximal compact subgroup, and assume that $\mathrm{rank } G = \mathrm {rank } K$. Let $\ g= l\oplus u$ be a $\theta$ stable parabolic subalgebra obtained by building $l$ from a subset of the compact simple roots and form $A_g(\lambda)$. Suppose $\Lambda=\lambda+2\delta( u\cap p)$ is $K$-dominant and the infinitesimal character, $\lambda+\delta$, of $A_{g}(\lambda)$ is nondominant due to a noncompact simple root. By interpreting these conditions on the level of Vogan diagrams, a conjecture by Knapp is (essentially) settled for the groups $G=SU(p,q),\, Sp(p,q)$, and $SO^*(2n)$, thereby determining the Langlands parameters of natural irreducible subquotient of $A_{ g}(\lambda)$. For the remaining classical groups, simple supplementary conditions are given under which the Langlands parameters may be determined.

scholarly journals HIGHER CODIMENSIONAL UEDA THEORY FOR A COMPACT SUBMANIFOLD WITH UNITARY FLAT NORMAL BUNDLE

2018 ◽  
Vol 238 ◽  
pp. 104-136
Author(s):  
TAKAYUKI KOIKE
Keyword(s):  
Line Bundle ◽  
Complex Manifold ◽  
Normal Bundle ◽  
Holomorphic Foliation ◽  
Compact Complex Manifold ◽  
Sufficient Condition ◽  
Existence Problem ◽  
Hermitian Metric ◽  
Flat Normal Bundle ◽  
Compact Submanifold

Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. Our interest is in a sort of the linearizability problem of a neighborhood of $Y$. As a higher codimensional generalization of Ueda’s result, we give a sufficient condition for the existence of a nonsingular holomorphic foliation on a neighborhood of $Y$ which includes $Y$ as a leaf with unitary-linear holonomy. We apply this result to the existence problem of a smooth Hermitian metric with semipositive curvature on a nef line bundle.

scholarly journals ON THE GEOMETRY OF THE SUPERMULTIPLET IN M-THEORY

2011 ◽  
Vol 08 (07) ◽  
pp. 1519-1551 ◽  
Author(s):  
HISHAM SATI
Keyword(s):  
Partition Function ◽  
Lie Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Internal Space ◽  
Bilinear Forms ◽  
M Theory ◽  
Kaluza Klein ◽  
String Theories

The massless supermultiplet of 11-dimensional supergravity can be generated from the decomposition of certain representation of the exceptional Lie group F4 into those of its maximal compact subgroup Spin(9). In an earlier paper, a dynamical Kaluza–Klein origin of this observation is proposed with internal space the Cayley plane, 𝕆P2, and topological aspects are explored. In this paper we consider the geometric aspects and characterize the corresponding forms which contribute to the action as well as cohomology classes, including torsion, which contribute to the partition function. This involves constructions with bilinear forms. The compatibility with various string theories are discussed, including reduction to loop bundles in ten dimensions.

scholarly journals Character formulas for discrete series on semisimple Lie groups

1976 ◽  
Vol 64 ◽  
pp. 47-61 ◽  
Author(s):  
Rebecca A. Herb
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Maximal Compact Subgroup ◽  
Discrete Series ◽  
Compact Subgroup ◽  
Semisimple Lie Groups ◽  
Analytic Group ◽  
Finite Center ◽  
Real Analytic ◽  
Analytic Subgroup

Let G be a connected, semisimple real Lie group with finite center, K a maximal compact subgroup of G. Assume rank G = rank K. Let be the Lie algebra of G, its complexification. If Gc is the simplyconnected complex analytic group corresponding to assume G is the real analytic subgroup of Gc corresponding to .

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