scholarly journals Convexity theorems for the gradient map on probability measures

2018 ◽  
Vol 5 (1) ◽  
pp. 133-145 ◽  
Author(s):  
Leonardo Biliotti ◽  
Alberto Raffero
Keyword(s):  
Lie Group ◽  
Kähler Manifold ◽  
Probability Measures ◽  
Kahler Manifold ◽  
Abelian Case ◽  
Real Submanifold ◽  
Reductive Lie Group

AbstractGiven a Kähler manifold (Z, J, ω) and a compact real submanifold M ⊂ Z, we study the properties of the gradient map associated with the action of a noncompact real reductive Lie group G on the space of probability measures on M. In particular, we prove convexity results for such map when G is Abelian and we investigate how to extend them to the non-Abelian case.

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 $\mu _{\mathfrak p}$ .

scholarly journals A Splitting Result for Real Submanifolds of a Kähler Manifold

2021 ◽  
Author(s):  
Leonardo Biliotti
Keyword(s):  
Lie Group ◽  
Kähler Manifold ◽  
Real Form ◽  
Real Submanifolds ◽  
Kahler Manifold ◽  
Reductive Lie Group ◽  
Simon Stevin ◽  
Connected Lie Group

AbstractLet $$(Z,\omega )$$ ( Z , ω ) be a connected Kähler manifold with an holomorphic action of the complex reductive Lie group $$U^\mathbb {C}$$ U C , where U is a compact connected Lie group acting in a hamiltonian fashion. Let G be a closed compatible Lie group of $$U^\mathbb {C}$$ U C and let M be a G-invariant connected submanifold of Z. Let $$x\in M$$ x ∈ M . If G is a real form of $$U^\mathbb {C}$$ U C , we investigate conditions such that $$G\cdot x$$ G · x compact implies $$U^\mathbb {C}\cdot x$$ U C · x is compact as well. The vice-versa is also investigated. We also characterize G-invariant real submanifolds such that the norm-square of the gradient map is constant. As an application, we prove a splitting result for real connected submanifolds of $$(Z,\omega )$$ ( Z , ω ) generalizing a result proved in Gori and Podestà (Ann Global Anal Geom 26: 315–318, 2004), see also Bedulli and Gori (Results Math 47: 194–198, 2005), Biliotti (Bull Belg Math Soc Simon Stevin 16: 107–116 2009).

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 FORMAL FROBENIUS MANIFOLD STRUCTURE ON EQUIVARIANT COHOMOLOGY

1999 ◽  
Vol 01 (04) ◽  
pp. 535-552 ◽  
Author(s):  
HUAI-DONG CAO ◽  
JIAN ZHOU
Keyword(s):  
Lie Group ◽  
Equivariant Cohomology ◽  
Kähler Manifold ◽  
Frobenius Manifold ◽  
Algebra Structure ◽  
Compact Lie Group ◽  
Hamiltonian Action ◽  
Manifold Structure ◽  
Kahler Manifold ◽  
Cartan Model

For a closed Kähler manifold with a Hamiltonian action of a connected compact Lie group by holomorphic isometries, we construct a formal Frobenius manifold structure on the equivariant cohomology by exploiting a natural DGBV algebra structure on the Cartan model.

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.

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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