arXiv subjects

Peter Heinzner

Publications and source records attributed to Peter Heinzner.

14 recordsLinked to original sources

arXiv2025

A structure theorem along fibers of extreme points of the momentum polytope

Let G be a complex reductive Lie group acting on a compact K\"ahler manifold X and assume that the action of a maximal compact subgroup K of G is Hamiltonian. For each extreme point of the convex hull of the momentum map image, there is an associated open dense subset of X, which is invariant under a parabolic subgroup Q of G. We prove a Q-equivariant product decomposition for the Q-action on this subset and discuss some applications of the result. We show a similar statement for real reductive subgroups of G for the restricted momentum map.

math.CV↗

arXiv2020

Invariant Kähler potentials and symplectic reduction

For a proper Hamiltonian action of a Lie group $G$ on a Kähler manifold $(X,ω)$ with momentum map $μ$ we show that the symplectic reduction $μ^{-1}(0)/G$ is a normal complex space. Every point in $μ^{-1}(0)$ has a $G$-stable open neighborhood on which $ω$ and $μ$ are given by a $G$-invariant Kähler potential. This is used to show that $μ^{-1}(0)/G$ is a Kähler space. Furthermore we examine the existence of potentials away from $μ^{-1}(0)$ with both positive and negative results.

math.SG↗

arXiv2020

Equivariant embeddings of strongly pseudoconvex Cauchy-Riemann manifolds

Let $X$ be a CR manifold with transversal, proper CR $G$-action. We show that $X/G$ is a complex space such that the quotient map is a CR map. Moreover the quotient is universal, i.e. every invariant CR map into a complex manifold factorises uniquely over a holomorphic map on $X/G$. We then use this result and complex geometry to proof an embedding theorem for (non-compact) strongly pseudoconvex CR manifolds with transversal $G \rtimes S^1$-action. The methods of the proof are applied to obtain a projective embedding theorem for compact CR manifolds.

math.CV↗

arXiv2014

Invariant convex sets in polar representations

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.

math.CV↗

arXiv2014

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 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 $μ_{\mathfrak{p}}(Y)$ is independent of the choice of the invariant Kähler form $ω$ in its cohomology class $[ω]$.

math.CV↗

arXiv2013

Polar orbitopes

We study polar orbitopes, i.e. convex hulls of orbits of a polar representation of a compact Lie group. The face structure is studied by means of the gradient momentum map and it is shown that every face is exposed and is again a polar orbitope. Up to conjugation the faces are completely determined by the momentum polytope. There is a tight relation with parabolic subgroups: the set of extreme points of a face is the closed orbit of a parabolic subgroup of G and for any parabolic subgroup the closed orbit is of this form.

math.RT↗

arXiv2013

Coadjoint orbitopes

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.

math.RT↗

arXiv2012

Stratifications with respect to actions of real reductive groups

We study the action of a real reductive group G on a real submanifold X of a K"ahler manifold Z. We suppose that the action of G extends holomorphically to an action of a complex reductive group and is Hamiltonian with respect to a compatible maximal compact subgroup of the complex reductive group. There is a corresponding gradient map obtained from a Cartan decomposition of G. We obtain a Morse like function on X. Associated to its critical points are various sets of semistable points which we study in great detail. In particular, we have G-stable submanifolds of X which are called pre-strata. In case that the gradient map is proper, the pre-strata form a decomposition of X and in case that 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 that X=Z is compact and the group itself is complex reductive.

math.CV↗

arXiv2007

Convexity properties of gradient maps

We consider the action of a real reductive group G on a Kaehler manifold Z which is the restriction of a holomorphic action of the complexified group G^C. We assume that the induced action of a compatible maximal compact subgroup U of G^C on Z is Hamiltonian. We have an associated gradient map obtained from a Cartan decomposition of G. For a G-stable subset Y of Z we consider convexity properties of the intersection of the image of Y under the gradient map with a closed Weyl chamber. Our main result is a Convexity Theorem for real semi-algebraic subsets Y of the projective space corresponding to a unitary representation of U.

math.CV↗

arXiv2007

Kaehlerian reduction in steps

We study Hamiltonian actions of compact Lie groups K on Kaehler 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 Kaehlerian reduction with respect to L is a stratified Hamiltonian Kaehler K^C/L^C-space whose Kaehlerian reduction with respect to K/L is naturally isomorphic to the Kaehlerian reduction of the original manifold with respect to K.

math.SG↗

arXiv2005

Semistable points with respect to real forms

We consider actions of real Lie subgroups G of complex reductive Lie groups on Kaehlerian spaces. Our main result is the openness of the set of semistable points with respect to a momentum map and the action of G.

math.CV↗

arXiv2003

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

Let C be the open upper light cone in $\mathbb R^{1+n}$ with respect to the Lorentz product. The connected linear Lorentz group SO$_\mathbb R(1,n)^0$ acts on C and therefore diagonally on the N-fold product $T^N$ where $T = \mathbb R^{1+n} + iC \subset \mathbb C^{1+n}$. We prove that the extended future tube SO$_\mathbb C(1,n) \cdot T^N$ is a domain of holomorphy.

math.CV↗

arXiv1997

The minimum principle from a Hamiltonian point of view

Let G be a complex Lie group, G_R a real form of G and X a G_R-stable domain of holomorphy in a complex G-manifold. If there is a G_R-invariant strictly plurisubharmonic function on X which has certain exhaustion properties, then we show that the extended domain G.X is also a domain of holomorphy. As an application we give a proof of the extended future tube conjecture. This is the assertion that G.X is a domain of holomorphy in the case where X is the N-fold product of the tube domain in C^4 over the positive light cone in R^4 and G is the connected complex Lorentz group acting diagonally.

dg-ga↗

arXiv1997

Projectivity of moment map quotients

Let G be a complex reductive group and K a maximal compact subgroup. If X is a smooth projective G-variety, with a fixed (not necessarily integral) K-invariant Kaehler form, then the K-action is Hamiltonian. Let M be the zero fiber of the corresponding moment map. It is well known that the quotient M/K is a complex space in a natural way. We prove that M/K is a projective variety. In particular, it follows that semistability with respect to a moment map is equivalent to semistability in the sense of Mumford.

dg-ga↗