Stability of measures on Kähler manifolds

Journal of Geometry and Physics

2020

Self Cite

Let G be a connected real reductive Lie group acting linearly on a finite dimensional vector space V over R. This action admits a Kempf-Ness function and so we have an associated gradient map. If G is Abelian we explicitly compute the image of G orbits under the gradient map, generalizing a result proved by Kac and Peterson [38]. A similar result is proved for the gradient map associated to the natural G action on P(V ). We also investigate the convex hull of the image of the gradient map restricted on the closure of G orbits. Finally, we give a new proof of the Hilbert-Mumford criterion for real reductive Lie groups avoiding any algebraic result.2010 Mathematics Subject Classification. 22E45,53D20; 14L24.

Section: This Equation Is Called the Cocycle Conditionmentioning

confidence: 83%

Section: Kempf-ness Functionsmentioning

confidence: 99%

See 1 more Smart Citation

Convexity properties of gradient maps associated to real reductive representations

Journal of Geometry and Physics

2020

Self Cite

“…The action of K extends to a holomorphic action of the complexification G := K C . Define F : P(X) → k * by the formula As explained in [7] this map is a momentum mapping for the action of K on P(X), in an appropriate sense.…”

Section: The Action On the Set Of Measuresmentioning

confidence: 99%

“…If ξ ∈ k and x ∈ X, then the limit lim t→+∞ exp(itξ) · x (1.1) always exists and defines a limit map, see e.g. [7,Prop. 5.18].…”

Section: Introductionmentioning

confidence: 99%

Meromorphic Limits of Automorphisms

Ghigi

2020

Transformation Groups

Let X be a compact complex manifold in the Fujiki class C . We study the compactification of Aut 0 (X) given by its closure in Barlet cycle space. The boundary points give rise to nondominant meromorphic self-maps of X. Moreover convergence in cycle space yields convergence of the corresponding meromorphic maps. There are analogous compactifications for reductive subgroups acting trivially on Alb X. If X is Kähler, these compactifications are projective. Finally we give applications to the action of Aut(X) on the set of probability measures on X. In particular we obtain an extension of Furstenberg lemma to manifolds in the class C .2010 Mathematics Subject Classification. Primary 32M05; Secondary 32M12.

Stability with respect to actions of real reductive Lie groups

Zedda

2017

Annali di Matematica

Self Cite

We give a systematic treatment of the stability theory for action of a real reductive Lie group G on a topological space. More precisely, we introduce an abstract setting for actions of non-compact real reductive Lie groups on topological spaces that admit functions similar to the Kempf-Ness function. The point of this construction is that one can characterize stability, semi-stability and polystability of a point by numerical criteria, that is in terms of a function called maximal weight. We apply this setting to the actions of a real non-compact reductive Lie group G on a real compact submanifold M of a Kähler manifold Z and to the action of G on measures of M .2010 Mathematics Subject Classification. 53D20; 14L24.