Peter Heinzner

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Papers by Peter Heinzner

Invariant convex sets in polar representations

Israel Journal of Mathematics, 2016

ABSTRACT We study a compact invariant convex set $E$ in a polar representation of a compact Lie g... more ABSTRACT We study a compact invariant convex set $E$ in a polar representation of a compact Lie group. Polar rapresentations are given by the adjoint action of $K$ on $\mathfrak{p}$, where $K$ is a maximal compact subgroup of a real semisimple Lie group $G$ with Lie algebra $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$. If $\mathfrak{a} \subset \mathfrak{p}$ is a maximal abelian subalgebra, then $P=E\cap \mathfrak{a}$ is a convex set in $\mathfrak{a}$. We prove that up to conjugacy the face structure of $E$ is completely determined by that of $P$ and that a face of $E$ is exposed if and only if the corresponding face of $P$ is exposed. We apply these results to the convex hull of the image of a restricted momentum map.

Komplexe fl�chen mit holomorphen vektorfeldern

Abh Math Sem Univ Hamburg, 1990

Semistable points with respect to real forms

Mathematische Annalen, 2007

We consider actions of real Lie subgroups G of complex reductive Lie groups on Kaehlerian spaces.... more

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A Remark on the Gradient Map

Documenta Mathematica, Feb 7, 2014

For a Hamiltonian action of a compact group $U$ of isometries on a compact K\"ahler manifold $Z$ ... more For a Hamiltonian action of a compact group $U$ of isometries on a compact K\"ahler manifold $Z$ and a compatible subgroup $G$ of $U^{\mathbb{C}}$, we prove that for any closed $G$--invariant subset $Y\subset Z$ the image of the gradient map $\mu_{\mathfrak{p}}(Y)$ is independent of the choice of the invariant K\"ahler form $\omega$ in its cohomology class $[\omega]$.

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Le principe d’Oka équivariant. (The equivariant Oka principle)

Comptes Rendus de l Académie des Sciences - Series I - Mathematics

Quotients with respect to holomorphic actions of reductive groups

Semistable quotients

Annali della Scuola normale superiore di Pisa, Classe di scienze

Projectivity of moment map quotients

OSAKA JOURNAL OF MATHEMATICS

Let be a complex reductive group acting algebraically on a complex projective variety . Given a p... more Let be a complex reductive group acting algebraically on a complex projective variety . Given a polarization of , i.e., an ample -line bundle Ä over , Mumford (see ) defined the notion of stability: A point Ü ¾ is said to be semistable with respect to Ä if and only if there exists Ñ ¾ N and an invariant section × : Ä Ñ such that ×(Ü) = 0. Let (Ä) denote the set of semistable points in , then there is a projective variety (Ä) and a -invariant surjective algebraic map : (Ä) (Ä) such that (i) is an affine map and

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The extended future tube conjecture for SO(1,n)

Let C be the open upper light cone in R 1+n with respect to the Lorentz product. The connected li... more Let C be the open upper light cone in R 1+n with respect to the Lorentz product. The connected linear Lorentz group SO R (1, n) 0 acts on C and therefore diagonally on the N -fold product T N where T = R 1+n + iC ⊂ C 1+n . We prove that the extended future tube SO C (1, n) · T N is a domain of holomorphy. * Supported by the SFB 237 of the DFG and the Ruth und Gerd Massenberg Stiftung

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Coadjoint orbitopes

OSAKA JOURNAL OF MATHEMATICS

We study coadjoint orbitopes, i.e. convex hulls of coadjoint orbits of a compact Lie group. We sh... more We study coadjoint orbitopes, i.e. convex hulls of coadjoint orbits of a compact Lie group. We show that all the faces of such an orbitope are exposed. The face structure is studied by means of the momentum map and it is shown that every face is again a coadjoint orbitope. Up to conjugation the faces are completely determined by the momentum polytope and can be described in a simple way in terms of root data. Finally we consider the complex geometry of the coadjoint orbit and we prove that the submanifolds of the orbit that are extreme sets of a face are exactly the closed orbits of parabolic subgroups.

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A remark on the gradient map

For a Hamiltonian action of a compact group U of isometries on a compact Kähler manifold Z and a ... more For a Hamiltonian action of a compact group U of isometries on a compact Kähler manifold Z and a compatible subgroup G of U C , we prove that for any closed G-invariant subset Y ⊂ Z the image of the gradient map µ p (Y ) is independent of the choice of the invariant Kähler form ω in its cohomology class [ω].

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Kählerian Reduction in Steps

Symmetry and Spaces, 2009

We study Hamiltonian actions of compact Lie groups K on Kähler manifolds which extend to a holomo... more We study Hamiltonian actions of compact Lie groups K on Kähler manifolds which extend to a holomorphic action of the complexified group K C . For a closed normal subgroup L of K we show that the Kählerian reduction with respect to L is a stratified Hamiltonian Kähler K C /L C -space whose Kählerian reduction with respect to K/L is naturally isomorphic to the Kählerian reduction of the original manifold with respect to K. * The first author is supported by a Promotionsstipendium of the Studienstiftung des deutschen Volkes and by SFB/TR 12 of the DFG.

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Critical Points of the Square of the Momentum Map

Global Aspects of Complex Geometry, 2006

ABSTRACT This paper which is made up of seven theorems, eight lemmas and propositions, nine remar... more ABSTRACT This paper which is made up of seven theorems, eight lemmas and propositions, nine remarks, eleven examples, two corollaries and one question deals with various properties of the momentum map associated with the Hamiltonian actions of the compact Lie group U on a Kählerian manifold M and the real form G of its complexification U ℂ . It is claimed that when G is a compatible Lie subgroup of U ℂ one can get a lot of information about the G-action on M. Details and complete proofs of various theorems, lemmas and propositions can be found in [P. Heinzner and G. Schwarz, Math. Ann. 337, No. 1, 197–232 (2007; Zbl 1110.32008)] and [P. Heinzner and H. Stötzel, Math. Ann. 338, No. 1, 1–9 (2007; Zbl 1129.32015)].

Actions of compact groups on coherent sheaves

Transformation Groups, 1999

Let K be a compact Lie group with complexification K C and X a reduced Stein space endowed with a... more Let K be a compact Lie group with complexification K C and X a reduced Stein space endowed with a continuous action of K by holomorphic transformations. In this set up there is a complex space X//K and a surjective K-invariant holomorphic map π: X → X//K such that the structure sheaf of X//K is the sheaf of invariants (π * O X ) K . Furthermore, the K-action on X can be complexified in the following sense: X can be realized as an open K-stable subset of a Stein space X c which is endowed with a holomorphic K C -action. Moreover, the inclusion X ⊂ X c is universal, i.e., every K-equivariant holomorphic map φ from X into a holomorphic K C -space Z extends uniquely and K C -equivariantly to a holomorphic map φ c : X c → Z (see [H]).

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Holomorphic actions on contractible domains without fixed points

by Peter Heinzner and Marco Abate

Mathematische Zeitschrift, 1992

Semistable points with respect to real forms

Mathematische Annalen, 2007

We consider actions of real Lie subgroups G of complex reductive Lie groups on Kählerian spaces. ... more

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Cartan decomposition of the moment map

Mathematische Annalen, 2006

We investigate a class of actions of real Lie groups on complex spaces. Using moment map techniqu... more We investigate a class of actions of real Lie groups on complex spaces. Using moment map techniques we establish the existence of a quotient and a version of Luna's slice theorem as well as a version of the Hilbert-Mumford criterion. A global slice theorem is proved for proper actions. We give new proofs of results of Mostow on decompositions of groups and homogeneous spaces. the irreducible components of X j , where d 1 = d 2 . By the argument above, if Y is an irreducible component of X 1 , then σ(Y ) is an irreducible component of X 2 . Since τ is invertible, the irreducible components of X 1 are mapped onto the irreducible components of X 2 . Hence σ(X 1 ) = X 2 .

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A global slice theorem for proper Hamiltonian actions

manuscripta mathematica, 1999

An analogue of a theorem of Abels [A] reducing the proper action of a noncompact Lie group to a c... more

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Spherical Stein spaces

manuscripta mathematica, 2004

Let X be an irreducible reduced complex space on which a connected compact Lie group K acts by ho... more Let X be an irreducible reduced complex space on which a connected compact Lie group K acts by holomorphic automorphisms. Let G be the complexification of K and g the Lie algebra of G. Following the theory of algebraic transformation groups, we call the complex space X spherical if X is normal and its tangent space at some point is generated by the vector fields from a Borel subalgebra b ⊂ g. We give several characterizations of spherical Stein spaces. In particular, we prove that a connected Stein manifold X is spherical if and only if the algebra of K-invariant differential operators on X is commutative.

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Kählerian extensions of the symplectic reduction

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

... DH Hamilton; 105 PDF. Kählerian extensions of the symplectic reduction. P. Heinzner; F. Loose... more

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