Actions Of Compact Lie Groups Research Articles

On gauge Poisson brackets with prescribed symmetry

Abstract In the context of averaging method, we describe a reconstruction of invariant connection-dependent Poisson structures from canonical actions of compact Lie groups on fibered phase spaces. Some symmetry properties of Wong's type equations are derived from the main results

A Multiplicative Tate Spectral Sequence for Compact Lie Group Actions

Given a compact Lie group G G and a commutative orthogonal ring spectrum R R such that R [ G ] ∗ = π ∗ ( R ∧ G + ) R[G]_* = \pi _*(R \wedge G_+) is finitely generated and projective over π ∗ ( R ) \pi _*(R) , we construct a multiplicative G G -Tate spectral sequence for each R R -module X X in orthogonal G G -spectra, with E 2 E^2 -page given by the Hopf algebra Tate cohomology of R [ G ] ∗ R[G]_* with coefficients in π ∗ ( X ) \pi _*(X) . Under mild hypotheses, such as X X being bounded below and the derived page R E ∞ RE^\infty vanishing, this spectral sequence converges strongly to the homotopy π ∗ ( X t G ) \pi _*(X^{tG}) of the G G -Tate construction X t G = [ E G ~ ∧ F ( E G + , X ) ] G X^{tG} = [\widetilde {EG}\wedge F(EG_+, X)]^G .

Vector Fields and Flows on Subcartesian Spaces

This paper is part of a series of papers on differential geometry of $C^\infty$-ringed spaces. In this paper, we study vector fields and their flows on a class of singular spaces. Our class includes arbitrary subspaces of manifolds, as well as symplectic and contact quotients by actions of compact Lie groups. We show that derivations of the $C^\infty$-ring of global smooth functions integrate to smooth flows.

Formality and finiteness in rational homotopy theory

We explore various formality and finiteness properties in the differential graded algebra models for the Sullivan algebra of piecewise polynomial rational forms on a space. The 1 -formality property of the space may be reinterpreted in terms of the filtered and graded formality properties of the Malcev Lie algebra of its fundamental group, while some of the finiteness properties of the space are mirrored in the finiteness properties of algebraic models associated with it. In turn, the formality and finiteness properties of algebraic models have strong implications on the geometry of the cohomology jump loci of the space. We illustrate the theory with examples drawn from complex algebraic geometry, actions of compact Lie groups, and 3 -dimensional manifolds.

Fixed point sets and the fundamental group II: Euler characteristics

For a finite group $G$ of not prime power order, Oliver showed that the obstruction for a finite CW-complex $F$ to be the fixed point set of a contractible finite $G$ -CW-complex is determined by the Euler characteristic $\chi (F)$ . (He also has similar results for compact Lie group actions.) We show that the analogous problem for $F$ to be the fixed point set of a finite $G$ -CW-complex of some given homotopy type is still determined by the Euler characteristic. Using trace maps on $K_0$ [2, 7, 18], we also see that there are interesting roles for the fundamental group and the component structure of the fixed point set.

K-Theory and Formality

Abstract We compare several notions of equivariant formality for complex K-theory with respect to a compact Lie group action, including surjectivity of the forgetful map and the weak equivariant formality of Harada–Landweber, and find all are equivalent under standard hypotheses. As a consequence, we present an expression for the equivariant K-theory of the isotropy action of $H$ on a homogeneous space $G/H$ in all the classical cases. The proofs involve mainly homological algebra and arguments with the Atiyah–Hirzebruch–Leray–Serre spectral sequence, but a more general result depends on a map of spectral sequences from Hodgkin’s Künneth spectral sequence in equivariant K-theory to that in Borel cohomology that seems not to have been otherwise defined. The hypotheses for the main structure result are analogous to a previously announced characterization of cohomological equivariant formality, first proved here, expanding on the results of Shiga and Takahashi.

Bismut–Cheeger Eta Form and Higher Spectral Flow

Abstract We prove the variation formula, the embedding formula, and the adiabatic limit formula for equivariant Bismut–Cheeger eta forms with a fiberwise compact Lie group action. We only need the condition that the kernels of corresponding fiberwise Dirac operators form vector bundles.

On cohomogeneity one Hermitian non-Kähler metrics

We investigate the geometry of Hermitian manifolds endowed with a compact Lie group action by holomorphic isometries with principal orbits of codimension one. In particular, we focus on a special class of these manifolds constructed by following Bérard-Bergery which includes, among the others, the holomorphic line bundles on $\mathbb {C}\mathbb {P}^{m-1}$, the linear Hopf manifolds and the Hirzebruch surfaces. We characterize their invariant special Hermitian metrics, such as balanced, Kähler-like, pluriclosed, locally conformally Kähler, Vaisman and Gauduchon. Furthermore, we construct new examples of cohomogeneity one Hermitian metrics solving the second-Chern–Einstein equation and the constant Chern-scalar curvature equation.

Visible actions of compact Lie groups on complex spherical varieties

With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the notion of visible actions on complex manifolds. He proved that for a Lie group $U$ the space of global sections of a $U$-equivariant holomorphic vector bundle $\mathcal{W}$ is multiplicity-free if the $U$-action on the base space $X$ is strongly visible and if the isotropy representations on the fibers are multiplicity-free under some compatibility condition on an anti-holomorphic diffeomorphism of $\mathcal{W}$. Especially in the case of the trivial line bundle, his theorem says that if a Lie group acts on the base space strongly visibly then the space of holomorphic functions is multiplicity-free. In short, the visibility is a geometric condition that assures the multiplicity-freeness property. In this article we consider the converse direction when $U$ is a compact real form of a connected complex reductive algebraic group $G$ and $X$ is an irreducible complex algebraic $G$-variety. In this setting the multiplicity-freeness property of the space of algebraic sections of arbitrary $G$-line bundle is equivalent to the sphericity of the $G$-action on $X$. Then our main result says that the multiplicity-freeness property implies the visibility, namely, if $X$ is a $G$-spherical variety then the $U$-action on $X$ is strongly visible.

Cohomogeneity one manifolds and homogeneous spaces of positive scalar curvature

We characterize cohomogeneity one manifolds and homogeneous spaces with a compact Lie group action admitting an invariant metric with positive scalar curvature.

Fredholm conditions for operators invariant with respect to compact Lie group actions

Let G be a compact Lie group acting smoothly on a smooth, compact manifold M, let P∈ψ m (M;E 0 ,E 1 ) be a G–invariant, classical pseudodifferential operator acting between sections of two G-equivariant vector bundles E i →M, i=0,1, and let α be an irreducible representation of the group G. Then P induces a map π α (P):H s (M;E 0 ) α →H s-m (M;E 1 ) α between the α-isotypical components. We prove that the map π α (P) is Fredholm if, and only if, P is transversally α-elliptic, a condition defined in terms of the principal symbol of P and the action of G on the vector bundles E i .

Equivariant eta forms and equivariant differential K-theory

In this paper, for a compact Lie group action, we prove the anomaly formula and the functoriality of the equivariant Bismut-Cheeger eta forms with perturbation operators when the equivariant family index vanishes. In order to prove them, we extend the Melrose-Piazza spectral section and its main properties to the equivariant case and introduce the equivariant version of the Dai-Zhang higher spectral flow for arbitrary-dimensional fibers. Using these results, we construct a new analytic model of the equivariant differential K-theory for compact manifolds when the group action has finite stabilizers only, which modifies the Bunke-Schick model of the differential K-theory. This model could also be regarded as an analytic model of the differential K-theory for compact orbifolds. Especially, we answer a question proposed by Bunke and Schick (2009) about the well-definedness of the push-forward map.

On Traces of Operators Associated with Actions of Compact Lie Groups

Given a pair $(M,X)$, where $X$ is a smooth submanifold in a closed smooth manifold $M$, we study the operation, which takes each operator $D$ on the ambient manifold to a certain operator on the submanifold. The latter operator is called the trace of $D$. More precisely, we study traces of operators, associated with actions of compact Lie groups on $M$. We show that traces of such operators are localized at special submanifolds in $X$ and study the structure of the traces on these submanifolds.

The asymptotic behavior of finite energy symplectic vortices with admissible metrics

Assume $(X, \omega)$ is a compact symplectic manifold with a Hamiltonian compact Lie group action and the zero in the Lie algebra is a regular value of the moment map $\mu$. We prove that a finite energy symplectic vortex exponentially converges to (un)twisted sectors of the symplectic reduction at cylinder ends whose metrics grow up at least cylindrically fast, without assuming the group action on the level set $\mu^{-1}(0)$ is free. It generalizes the corresponding results by Ziltener [23, 24] under the free action assumption. The result of this paper is the first step in setting up the quotient morphism moduli space induced by the authors in [6]. Necessary preparations in understanding the structure of such moduli spaces are also introduced here. The quotient morphism constructed in [6] is a part of the project on the quantum Kirwan morphism by the authors (see [3, 4, 5]).

Fibrations and stability for compact group actions on manifolds with local bounded Ricci covering geometry

In the study of the collapsed manifolds with bounded sectional curvature, the following two results provide basic tools: a (singular) fibration theorem by K. Fukaya [J. Differential Geom., 1987, 25(1): 139–156] and J. Cheeger, K. Fukaya, and M. Gromov [J. Amer. Math. Soc., 1992, 5(2): 327–372], and the stability for isometric compact Lie group actions on manifolds by R. S. Palais [Bull. Amer. Math. Soc., 1961, 67(4): 362–364] and K. Grove and H. Karcher [Math. Z., 1973, 132: 11–20]. The main results in this paper (partially) generalize the two results to manifolds with local bounded Ricci covering geometry.

Multisymplectic actions of compact Lie groups on spheres

We investigate the existence of homotopy comoment maps (comoments) for high-dimensional spheres seen as multisymplectic manifolds. Especially, we solve the existence problem for compact effective group actions on spheres and provide explicit constructions for such comoments in interesting particular cases.

Actions of some simple compact Lie groups on themselves

Let G be a compact connected simple Lie group acting non-transitively, non-trivially on itself. Hsiang (Cohomology theory of topological transformation groups, (1975) (New York: Springer)) conjectured that the principal isotropy subgroup type must be the maximal torus and the action must be cohomologically similar to the adjoint action and the orbit space must be a simplex. But Bredon (Bull AMS 83(4) (1977) 711–718) found a simple counterexample, where the principal isotropy subgroup is not a maximal torus and which has no fixed point. In this work, we prove that if $$SO(n)$$ , ( $$n\ge 34$$ ) or $$SU(3)$$ acts smoothly (and nontrivially) on itself with non-empty fixed point set, then the principal isotropy subgroups are maximal tori.

FINITE FLAT SPACES

We say that a finite metric space $X$ can be embedded almost isometrically into a class of metric spaces $C$, if for every $\epsilon > 0$ there exists an embedding of $X$ into one of the elements of $C$ with the bi-Lipschitz distortion less then $1 + \epsilon$. We show that almost isometric embeddability conditions are equal for following classes of spaces (a) Quotients of Euclidean spaces by isometric actions of finite groups, (b) $L_2$-Wasserstein spaces over Euclidean spaces, (c) Compact flat manifolds, (d) Compact flat orbifolds, (e) Quotients of connected compact bi-invariant Lie groups by isometric actions of compact Lie groups. (This one is the most surprising.) We call spaces which satisfy this conditions finite flat spaces. The question about synthetic definition naturally arises. Since Markov type constants depend only on finite subsets we can conclude that connected compact bi-invariant Lie groups and their quotients have Markov type $2$ with constant $1$.

Twisted equivariant K-theory of compact Lie group actions with maximal rank isotropy

We consider twisted equivariant K--theory for actions of a compact Lie group $G$ on a space $X$ where all the isotropy subgroups are connected and of maximal rank. We show that the associated rational spectral sequence \`a la Segal has a simple $E_2$--term expressible as invariants under the Weyl group of $G$. Namely, if $T$ is a maximal torus of $G$, they are invariants of the $\pi_1(X^T)$-equivariant Bredon cohomology of the universal cover of $X^T$ with suitable coefficients. In the case of the inertia stack $\Lambda Y$ this term can be expressed using the cohomology of $Y^T$ and algebraic invariants associated to the Lie group and the twisting. A number of calculations are provided. In particular, we recover the rational Verlinde algebra when $Y=\{*\}$.

The Topology of Compact Lie Group Actions Through the Lens of Finite Models

AbstractGiven a compact, connected Lie group K, we use principal K-bundles to construct manifolds with prescribed finite-dimensional algebraic models. Conversely, let M be a compact, connected, smooth manifold, which supports an almost free K-action. Under a partial formality assumption on the orbit space and a regularity assumption on the characteristic classes of the action, we describe an algebraic model for M with commensurate finiteness and partial formality properties. The existence of such a model has various implications on the structure of the cohomology jump loci of M and of the representation varieties of π1(M). As an application, we show that compact Sasakian manifolds of dimension 2n + 1 are (n − 1)-formal, and that their fundamental groups are filtered-formal. Further applications to the study of weighted-homogeneous isolated surface singularities are also given.