scholarly journals Generalized quivers, orthogonal and symplectic representations, and Hitchin–Kobayashi correspondences

2019 ◽  
Vol 30 (03) ◽  
pp. 1850085
Author(s):  
Artur de Araujo
Keyword(s):  
Reductive Group ◽  
Kähler Manifold ◽  
Kahler Manifold ◽  
Quiver Bundles ◽  
Symplectic Representations

We review the theory of quiver bundles over a Kähler manifold, and then introduce the concept of generalized quiver bundles for an arbitrary reductive group [Formula: see text]. We first study the case when [Formula: see text] or [Formula: see text], interpreting them as orthogonal (respectively symplectic) bundle representations of the symmetric quivers introduced by Derksen–Weyman. We also study supermixed quivers, which simultaneously involve both orthogonal and symplectic symmetries. In particular, we completely characterize the polystable forms of such representations. Finally, we discuss Hitchin–Kobayashi correspondences for these objects.

scholarly journals Stratifications with respect to actions of real reductive groups

2008 ◽  
Vol 144 (1) ◽  
pp. 163-185 ◽  
Author(s):  
Peter Heinzner ◽  
Gerald W. Schwarz ◽  
Henrik Stötzel
Keyword(s):  
Critical Points ◽  
Maximal Compact Subgroup ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Kähler Manifold ◽  
Reductive Groups ◽  
Cartan Decomposition ◽  
Kahler Manifold ◽  
Real Submanifold

AbstractWe study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of G extends holomorphically to an action of the complexified group $G^{\mathbb {C}}$ and that with respect to a compatible maximal compact subgroup U of $G^{\mathbb {C}}$ the action on Z is Hamiltonian. There is a corresponding gradient map $\mu _{\mathfrak {p}}\colon X\to \mathfrak {p}^*$ where $\mathfrak {g}=\mathfrak {k}\oplus \mathfrak {p}$ is a Cartan decomposition of $\mathfrak {g}$. We obtain a Morse-like function $\eta _{\mathfrak {p}}:=\Vert \mu _{\mathfrak {p}}\Vert ^2$ on X. Associated with critical points of $\eta _{\mathfrak {p}}$ are various sets of semistable points which we study in great detail. In particular, we have G-stable submanifolds Sβ of X which are called pre-strata. In cases where $\mu _{\mathfrak {p}}$ is proper, the pre-strata form a decomposition of X and in cases where X is compact they are the strata of a Morse-type stratification of X. Our results are generalizations of results of Kirwan obtained in the case where $G=U^{\mathbb {C}}$ and X=Z is compact.

scholarly journals COMPACT ORBITS OF PARABOLIC SUBGROUPS

2021 ◽  
pp. 1-9
Author(s):  
LEONARDO BILIOTTI ◽  
OLUWAGBENGA JOSHUA WINDARE
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Reductive Group ◽  
Kähler Manifold ◽  
Parabolic Subgroups ◽  
Cartan Decomposition ◽  
Kahler Manifold ◽  
Real Submanifold ◽  
Connected Lie Group

Abstract We study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of a compact connected Lie group U with Lie algebra $\mathfrak {u}$ extends holomorphically to an action of the complexified group $U^{\mathbb {C}}$ and that the U-action on Z is Hamiltonian. If $G\subset U^{\mathbb {C}}$ is compatible, there exists a gradient map $\mu _{\mathfrak p}:X \longrightarrow \mathfrak p$ where $\mathfrak g=\mathfrak k \oplus \mathfrak p$ is a Cartan decomposition of $\mathfrak g$ . In this paper, we describe compact orbits of parabolic subgroups of G in terms of the gradient map $\mu _{\mathfrak p}$ .

Homogeneous locally conformally Kähler and Sasaki manifolds

2015 ◽  
Vol 26 (06) ◽  
pp. 1541001 ◽  
Author(s):  
D. V. Alekseevsky ◽  
V. Cortés ◽  
K. Hasegawa ◽  
Y. Kamishima
Keyword(s):  
Lie Groups ◽  
Reductive Group ◽  
Kähler Manifold ◽  
Symplectic Manifolds ◽  
Isotropy Group ◽  
Kahler Manifold ◽  
Locally Conformally Kähler ◽  
Reductive Lie Groups ◽  
Locally Conformally Symplectic ◽  
Sasaki Manifolds

We prove various classification results for homogeneous locally conformally symplectic manifolds. In particular, we show that a homogeneous locally conformally Kähler manifold of a reductive group is of Vaisman type if the normalizer of the isotropy group is compact. We also show that such a result does not hold in the case of non-compact normalizer and determine all left-invariant lcK structures on reductive Lie groups.

Approximate Hitchin–Kobayashi correspondence for Higgs G-bundles

2014 ◽  
Vol 11 (07) ◽  
pp. 1460015 ◽  
Author(s):  
Ugo Bruzzo ◽  
Beatriz Graña Otero
Keyword(s):  
Reductive Group ◽  
Kähler Manifold ◽  
Higgs Bundle ◽  
Structure Group ◽  
Higgs Bundles ◽  
Yang Mills ◽  
Kahler Manifold ◽  
Group A ◽  
Compact Kähler Manifold

We announce a result about the extension of the Hitchin–Kobayashi correspondence to principal Higgs bundles. A principal Higgs bundle on a compact Kähler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian–Yang–Mills structure.

scholarly journals Relative non-pluripolar product of currents

2021 ◽  
Author(s):  
Duc-Viet Vu
Keyword(s):  
Kähler Manifold ◽  
Monotonicity Property ◽  
Necessary Condition ◽  
Positive Current ◽  
Lelong Numbers ◽  
Kahler Manifold ◽  
Closed Positive Current ◽  
Positive Currents ◽  
Compact Kähler Manifold

AbstractLet X be a compact Kähler manifold. Let $$T_1, \ldots , T_m$$ T 1 , … , T m be closed positive currents of bi-degree (1, 1) on X and T an arbitrary closed positive current on X. We introduce the non-pluripolar product relative to T of $$T_1, \ldots , T_m$$ T 1 , … , T m . We recover the well-known non-pluripolar product of $$T_1, \ldots , T_m$$ T 1 , … , T m when T is the current of integration along X. Our main results are a monotonicity property of relative non-pluripolar products, a necessary condition for currents to be of relative full mass intersection in terms of Lelong numbers, and the convexity of weighted classes of currents of relative full mass intersection. The former two results are new even when T is the current of integration along X.

scholarly journals Traceless component of the conformal curvature tensor in Kähler manifold

2006 ◽  
Vol 56 (3) ◽  
pp. 857-874 ◽  
Author(s):  
S. Funabashi ◽  
H. S. Kim ◽  
Y.-M. Kim ◽  
J. S. Pak
Keyword(s):  
Curvature Tensor ◽  
Kähler Manifold ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Kahler Manifold

A POSITIVITY PROPERTY FOR FOLIATIONS ON COMPACT KÄHLER MANIFOLDS

2006 ◽  
Vol 17 (01) ◽  
pp. 35-43 ◽  
Author(s):  
MARCO BRUNELLA
Keyword(s):  
Kähler Manifold ◽  
Rational Curves ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Canonical Bundle ◽  
Positivity Property ◽  
Kahler Manifold ◽  
Compact Kähler Manifold

We prove that the canonical bundle of a foliation by curves on a compact Kähler manifold is pseudoeffective, unless the foliation is a (special) foliation by rational curves.

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