Consider a Hamiltonian action of a compact Lie group K on a compact symplectic manifold. We find descriptions of the kernel of the Kirwan map corresponding to a regular value of the moment map $κ_K$. We start with the case when K is a torus T: we determine the kernel of the equivariant Kirwan map (defined by Goldin in [Go]) corresponding to a generic circle S in T, and show how to recover from this the kernel of $κ_T$, as described by Tolman and Weitsman.
(In the situation when the fixed point set of the torus action is finite, similar results have been obtained in our previous papers [Je], [Je-Ma]). For a compact nonabelian Lie group K we will use the ``non-abelian localization formula'' of [Je-Ki1] and [Je-Ki2] to establish relationships -- some of them obtained by Tolman and Weitsman in [To-We] -- between $\ker(κ_K)$ and $\ker(κ_T)$, where T is a maximal torus in K.
An Appendix generalizes Theorem 1.8 to the case of singular values of $κ_T$.
Using the notion of equivariant Kirwan map, as defined by Goldin, we prove that -- in the case of Hamiltonian torus actions with isolated fixed points -- Tolman and Weitsman's description of the kernel of the Kirwan map can be deduced directly from the residue theorem of Jeffrey and Kirwan. A characterization of the kernel of the Kirwan map in terms of residues of one variable (i.e. associated to Hamiltonian circle actions) is obtained.
Consider the holomorphic Hamiltonian action of a compact Lie group $K$ on a compact Kähler manifold $M$ with a moment map $Φ: M\rightarrow \mathfrak{k}^*$. Assume that $0$ is a regular value of the moment map. Weitsman raised the question of what we can say about the cohomology of the Kähler quotient $M_0:=Φ^{-1}(0)/K$ if all the ordinary cohomology of $M$ is of type $(p, p)$. In this paper, using the Cartan-Chern-Weil theory we show that in the above context there is a natural surjective Kirwan map from an equivariant version of the Dolbeault cohomology of $M$ onto the Dolbeault cohomology of…
We generalize the Jeffrey-Kirwan localization theorem for non-compact symplectic and hyperKahler quotients. Similarly to the circle compact integration of Hausel and Proudfoot we define equivariant integrals on non-compact manifolds using the Atiyah-Bott-Berline-Vergne localization formula as formal definition. We introduce a so called equivariant Jeffrey-Kirwan residue and we show that it shares similar properties as the usual one. Our localization formula has the same structure as the usual Jeffrey-Kirwan formula, but it uses formal integration and equivariant residue. We also give a version…
This paper uses Morse-theoretic techniques to compute the equivariant Betti numbers of the space of semistable rank two degree zero Higgs bundles over a compact Riemann surface, a method in the spirit of Atiyah and Bott's original approach for semistable holomorphic bundles. This leads to a natural proof that the hyperkähler Kirwan map is surjective for the non-fixed determinant case.
Let $T$ be a compact torus and $(M,ω)$ a Hamiltonian $T$-space. We give a new proof of the $K$-theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry by using the equivariant version of the Kirwan map introduced in one of R. Goldin's papers. We compute the kernel of this equivariant Kirwan map, and hence give a computation of the kernel of the Kirwan map. As an application, we find the presentation of the kernel of the Kirwan map for the $T$-equivariant $K$-theory of flag varieties $G/T$ where $G$ is a compact, connected and simply-connected Lie group. In the last section…
Let G be a compact Lie group and LG its associated loop group. The main result of this manuscript is a surjectivity theorem from the equivariant K-theory of a Hamiltonian LG-space onto the integral K-theory of its Hamiltonian LG-quotient. Our result is a K-theoretic analogue of previous work in rational Borel-equivariant cohomology of Bott, Tolman, and Weitsman. Our proof techniques differ from that of Bott, Tolman, and Weitsman in that they explicitly use the Borel construction, which we do not have at our disposal in equivariant K-theory; we instead directly construct G-equivariant homotopy …
Let $M$ be a smooth manifold and $G$ a compact connected Lie group acting on $M$ by isometries. In this paper, we study the equivariant cohomology of ${\bf X}=T^\ast M$, and relate it to the cohomology of the Marsden-Weinstein reduced space via certain residue formulae. In case that $\bf X$ is a compact symplectic manifold with a Hamiltonian $G$-action, similar residue formulae were derived by Jeffrey, Kirwan et al.
Let G be a compact torus acting on a compact symplectic manifold M in a Hamiltonian fashion, and T a subtorus of G. We prove that the kernel of $κ:H_G^*(M)\to H^*(M//G)$ is generated by a small number of classes $α\in H_G^*(M)$ satisfying very explicit restriction properties. Our main tool is the equivariant Kirwan map, a natural map from the G-equivariant cohomology of M to the G/T-equivariant cohomology of the symplectic reduction of M by T. We show this map is surjective. This is an equivariant version of the well-known result that the (nonequivariant) Kirwan map $κ:H_G^*(M)\to H^*(M//G)$ i…
Consider the Hamiltonian action of a torus on a compact twisted generalized complex manifold $M$. We first observe that Kirwan injectivity and surjectivity hold for ordinary equivariant cohomology in this setting. Then we prove that these two results hold for the twisted equivariant cohomology as well.
In this thesis we study the topology and geometry of hyperkähler quotients, as well as some related non-compact Kähler quotients, from the point of view of Hamiltonian group actions. The main technical tool we employ is Morse theory with moment maps. We prove a Lojasiewicz inequality which permits the use of Morse theory in the non-compact setting. We use this to deduce Kirwan surjectivity for an interesting class of non-compact quotients, and obtain a new proof of hyperkähler Kirwan surjectivity for hypertoric varieties. We then turn our attention to quiver varieties, obtaining an explicit in…
Consider a Hamiltonian action of a compact Lie group on a symplectic manifold which has the strong Lefschetz property. We establish an equivariant version of the Merkulov-Guillemin $dδ$-lemma and an improved version of the Kirwan-Ginzburg equivariant formality theorem, which says that every cohomology class has a \emph{canonical} equivariant extension.
We study a class of group actions on hyperkähler manifolds which we call actions of linear type. If $M$ is a hyperkähler manifold possessing such a $G$-action, the hyperkähler Kirwan map is surjective if and only if the natural restriction $H^\ast(M / G) \to H^\ast(M / G)$ is surjective. We prove that this restriction is an isomorphism below middle degree and an injection in middle degree. As a consequence, the hyperkähler Kirwan map is surjective except possibly in middle degree, and its kernel may be determined from the kernel of the ordinary Kirwan map. These results apply in particular to …
For a finite-dimensional (but possibly noncompact) symplectic manifold with a compact group acting with a proper moment map, we show that the square of the moment map is an equivariantly perfect Morse function in the sense of Kirwan, and that the set of critical points of the square of the moment map is a countable discrete union of compact sets. We show that certain integrals of equivariant cohomology classes localizes as a sum of contributions from these compact critical sets, and we bound the contribution from each critical set. In the case (1) that the contribution from higher critical set…
Let $M$ be a proper Hamiltonian $K$-space with proper moment map $μ$. The symplectic quotient $X=μ^{-1}(0)/K$ is in general a singular stratified space. In this paper we first generalize the Kirwan map to this symplectic setting which maps the $K$ equivariant cohomology of $μ^{-1}(0)$ to the middle perversity intersection cohomology $IH^*(X)$. Next, we show that there is a natural right inverse of the Kirwan map, under an assumption coming from the cosupport axiom for the intersection homology sheaf. Furthermore, we can identify the subspace of $H^*_K(μ^{-1}(0))$ which corresponds to $IH^*(X)$…
Lisa Jeffrey and Frances Kirwan developed an integration theory for symplectic reductions. That is, given a symplectic manifold with symplectic group action, they developed a way of pulling the integration of forms on the reduction back to an integration of group-equivariant forms on the original space. We seek an analogue of the symplectic integration formula as developed by for the hyperKahler case. This is almost straightforward, but we have to overcome such obstacles as the lack of a hyper-Darboux theorem and the lack of compactness in the case of hyperKahler reduction.
We give a simple direct proof (for the case of Hamiltonian circle actions with isolated fixed points) that Tolman and Weitsman's description of the kernel of the Kirwan map (in other words the sum of those equivariant cohomology classes vanishing on one side of a collection of hyperplanes) is equivalent to the characterization of this kernel given by the residue theorem of Jeffrey and Kirwan.
We develop a Chern-Weil theory for compact Lie group action whose generic stabilizers are finite in the framework of equivariant cohomology. This provides a method of changing an equivariant closed form within its cohomological class to a form more suitable to yield localization results. This work is motivated by our work on reproving wall crossing formulas in Seiberg-Witten theory, where the Lie group is the circle. As applications, we derive two localization formulas of Kalkman type for G = SU(2) or SO(3)-actions on compact manifolds with boundary. One of the formulas is then used to yield a…
Suppose given a linearized action on a polarized complex projective manifold (M,L), and assume that the stable locus is non-empty. We study the leading asymptotics of the dimension of the equivariant summands appearing in the space of global sections of high powers of L. We use Kirwan resolutions and an elementary algebro-geometric argument to reduce the problem to the case of non-singular actions (that is, actions for which the stable and semistable loci coincide and are non-empty).
We construct a morphism from the equivariant Fukaya algebra of a Lagrangian brane in the zero level set of a moment map of a Hamiltonian action to the Fukaya algebra of the quotient brane. This morphism induces a map between Maurer-Cartan solution spaces, and intertwines the disk potentials. As an application, we show under some technical hypotheses that weak unobstructedness of an invariant Lagrangian brane implies weak unobstructedness of its quotient. For semi-Fano toric manifolds we give a different proof of the open mirror theorem of Chan-Lau-Leung-Tseng by showing that the potential of a…
Let $M$ be a symplectic manifold and $G$ a connected, compact Lie group acting on $M$ in a Hamiltonian way. In this paper, we study the equivariant cohomology of $M$ represented by basic differential forms, and relate it to the cohomology of the Marsden-Weinstein reduced space via certain residue formulae using resolution of singularities and the stationary phase principle. In case that $ M $ is a compact, symplectic manifold or the co-tangent bundle of a $G$-manifold, similar residue formulae were derived by Jeffrey, Kirwan et al. for general equivariantly closed forms and by Ramacher for bas…
Consider a Hamiltonian action of a compact connected Lie group on a symplectic manifold $(M,ω)$. Conjecturally, under suitable assumptions there exists a morphism of cohomological field theories from the equivariant Gromov-Witten theory of $(M,ω)$ to the Gromov-Witten theory of the symplectic quotient. The morphism should be a deformation of the Kirwan map. The idea, due to D. A. Salamon, is to define such a deformation by counting gauge equivalence classes of symplectic vortices over the complex plane $C$. The present memoir is part of a project whose goal is to make this definition rigorous.…
Given a compact symplectic manifold M with the Hamiltonian action of a torus T, let zero be a regular value of the moment map, and M_0 the symplectic reduction at zero. Denote by κ_0 the Kirwan map H^*_T(M)-> H^*(M_0). For an equivariant cohomology class η\in H^*_T(M) we present new localization formulas which express \int_{M_0} κ_0(η) as sums of certain integrals over the connected components of the fixed point set M^T. To produce such a formula we apply a residue operation to the Atiyah-Bott-Berline-Vergne localization formula for an equivariant form on the symplectic cut of M with respect t…
Motivated by Nekrasov's instanton counting, we discuss a method for calculating equivariant volumes of non-compact quotients in symplectic and hyper-Kähler geometry by means of the Jeffrey-Kirwan residue-formula of non-abelian localization. In order to overcome the non-compactness, we use varying symplectic cuts to reduce the problem to a compact setting, and study what happens in the limit that recovers the original problem. We implement this method for the ADHM construction of the moduli spaces of framed Yang-Mills instantons on $\R^{4}$ and rederive the formulas for the equivariant volumes …
It has been shown recently by Kapustin and Tomasiello that the mathematical notion of Hamiltonian actions on twisted generalized Kähler manifolds is in perfect agreement with the physical notion of general $(2,2)$ gauged sigma models with three-form fluxes. In this article, we study the twisted equivariant cohomology theory of Hamiltonian actions on $H$-twisted generalized complex manifolds. If the manifold satisfies the $\bar{\partial}\partial$-lemma, we establish the equivariant formality theorem. If in addition, the manifold satisfies the generalized Kähler condition, we prove the Kirwan in…
Suppose that an algebraic torus $G$ acts algebraically on a projective manifold $X$ with generically trivial stabilizers. Then the Zariski closure of the set of pairs $\{(x,y)\in X\times X\mid y=gx \text{for some}g\in G\}$ defines a nonzero equivariant cohomology class $[Δ_G]\in H^*_{G\times G}(X\times X)$. We give an analogue of this construction in the case where $X$ is a compact symplectic manifold endowed with a hamiltonian action of a torus, whose complexification plays the role of $G$. We also prove that the Kirwan map sends the class $[Δ_G]$ to the class of the diagonal in each symplect…
Let $M$ be a symplectic manifold equipped with a Hamiltonian action of a torus $T$. Let $F$ denote the fixed point set of the $T$-action and let $i:F\hookrightarrow M$ denote the inclusion. By a theorem of F. Kirwan \cite{K} the induced map $i^*:H_T^*(M) \to H_T^*(F)$ in equivariant cohomology is an injection. We give a simple proof of a formula of Goresky-Kottwitz-MacPherson \cite{GKM} for the image of the map $i^*$.
Let G be a compact connected Lie group, and (M,ω) a Hamiltonian G-space with proper moment map μ. We give a surjectivity result which expresses the K-theory of the symplectic quotient M//G in terms of the equivariant K-theory of the original manifold M, under certain technical conditions on μ. This result is a natural K-theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the K-theoretic Atiyah-Bott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian G-spaces. We discuss this lemma in detail and highlight the diffe…
In this article we are concerned with how to compute the cohomology ring of a symplectic quotient by a circle action using the information we have about the cohomology of the original manifold and some data at the fixed point set of the action. Our method is based on the Tolman-Weitsman theorem which gives a characterization of the kernel of the Kirwan map. First we compute a generating set for the kernel of the Kirwan map for the case of product of compact connected manifolds such that the cohomology ring of each of them is generated by a degree two class. We assume the fixed point set is iso…
A polynomial assignment for a continuous action of a compact torus $T$ on a topological space $X$ assigns to each $p\in X$ a polynomial function on the Lie algebra of the isotropy group at $p$ in such a way that a certain compatibility condition is satisfied. The space ${\mathcal{A}}_T(X)$ of all polynomial assignments has a natural structure of an algebra over the polynomial ring of ${\rm Lie}(T)$. It is an equivariant homotopy invariant, canonically related to the equivariant cohomology algebra. In this paper we prove various properties of ${\mathcal{A}}_T(X)$ such as Borel localization, a C…
In this note we prove the following theorem: Let $G$ be a compact Lie group acting on a compact symplectic manifold $M$ in a Hamiltonian fashion. If $L$ is an $l$-dimensional closed invariant submanifold of $M$, on which the $G$-action is locally free then the fundamental class $[L]$ is trivial in $H_l(M,{\mathbb Q})$. We also prove similar results for lower homology groups of $L$, in case the group $G$ is a finite product of copies of $S^1$ and SU(2). The key ingredients of the proofs are Kirwan's theorem that Hamiltonian spaces are equivariantly formal and symplectic reduction.
Suppose given an Hamiltonian action of a compact semisimple Lie group on a polarized complex projective manifold $(M,L)$. We study by means of microlocal techniques the local and global asymptotic behaviour of linear series on $M$ defined in terms of the action and the irreducible representations of $G$.
Suppose that a compact and connected Lie group $G$ acts on a complex Hodge manifold $M$ in a holomorphic and Hamiltonian manner, and that the action linearizes to a positive holomorphic line bundle $A$ on $M$. Then there is an induced unitary representation on the associated Hardy space and, if the moment map of the action is nowhere vanishing, the corresponding isotypical components are all finite dimensional. We study the asymptotic concentration behavior of the corresponding equivariant Szegö kernels near certain loci defined by the moment map.
Let $M$ be a compact, connected symplectic manifold with a Hamiltonian action of a compact $n$-dimensional torus $T$. Suppose that $M$ is equipped with an anti-symplectic involution $σ$ compatible with the $T$-action. The real locus of $M$ is the fixed point set $M^σ$ of $σ$. Duistermaat introduced real loci, and extended several theorems of symplectic geometry to real loci. In this paper, we extend another classical result of symplectic geometry to real loci: the Kirwan surjectivity theorem. In addition, we compute the kernel of the real Kirwan map. These results are direct consequences of te…
The space of Fredholm operators of fixed index is stratified by submanifolds according to the dimension of the kernel. Geometric considerations often lead to questions about the intersections of concrete families of elliptic operators with these submanifolds: are the intersections non-empty? are they smooth? what are their codimensions? The purpose of this article is to develop tools to address these questions in equivariant situations. An important motivation for this work are transversality questions for multiple covers of $J$-holomorphic maps. As an application, we use our framework to give…
Suppose that the compact and connected Lie group G acts holomorphically on the irreducible complex projective manifold M, and that the action linearizes to the Hermitian ample line bundle L on M. Assume that 0 is a regular value of the associated moment map. The spaces of global holomorphic sections of powers of L may be decomposed over the finite dimensional irreducible representations of G. In this paper, we study how the holomorphic sections in each equivariant piece asymptotically concentrate along the zero locus of the moment map. In the special case where G acts freely on the zero locus …
Let $M$ be complex projective manifold, and $A$ a positive line bundle on it. Assume that $SU(2)$ acts on $M$ in a Hamiltonian manner, with nowhere vanishing moment map, and that this action linearizes to $A$. Then there is an associated unitary representation of $G$ on the associated algebro-geometric Hardy space, and the isotypical components are all finite dimensional. We consider the local and global asymptotic properties the equivariant projector associated to a weight $k \, \boldsymbol{ ν}$, when $\boldsymbol{ ν}$ is fixed and $k\rightarrow +\infty$.
Let $M$ be complex projective manifold and $A$ a positive line bundle on it. Assume that a compact and connected Lie group $G$ acts on $M$ in a Hamiltonian and holomorphic manner and that this action linearizes to $A$. Then, there is an associated unitary representation of $G$ on the associated algebro-geometric Hardy space $H(X)$. The standard circle action on $H(X)$ commutes with the action of $G$ and thus one has a decompositions labeled by $(k\,\boldsymbolν,\,k)$, where $k\in\mathbb{Z}$ and $\boldsymbol{ ν}\in \hat{G}$. We consider the local and global asymptotic properties of the correspo…
Let $G/P$ be a generalized flag variety, where $G$ is a complex semisimple connected Lie group and $P\subset G$ a parabolic subgroup. Let also $X\subset G/P$ be a Schubert variety. We consider the canonical embedding of $X$ into a projective space, which is obtained by identifying $G/P$ with a coadjoint orbit of the compact Lie group $K$, where $G=K^{\mathbb C}$. The maximal torus $T$ of $K$ acts linearly on the projective space and it leaves $X$ invariant: let $Ψ: X \to {\rm Lie}(T)^*$ be the restriction of the moment map relative to the Fubini-Study symplectic form. By a theorem of Atiyah, $…
The subject of this dissertation is the Gysin homomorphism in equivariant cohomology for spaces with torus action. We consider spaces which are quotients of classical semisimple complex linear algebraic groups by a parabolic subgroup with the natural action of a maximal torus, the so-called partial flag varieties. We derive formulas for the Gysin homomorphism for the projection to a point, of the form \[\int_X α= Res_{\mathbf{z}=\infty} \mathcal{Z}(\mathbf{z}, \mathbf{t}) \cdot α(\mathbf{t}),\] for a certain residue operation and a map $\mathcal{Z}(\mathbf{z}, \mathbf{t})$, associated to the r…
Let $γ$ be an automorphism of a polarized complex projective manifold $(M,L)$. Then $γ$ induces an automorphism $γ_k$ of the space of global holomorphic sections of the $k$-th tensor power of $L$, for every $k=1,2,...$; for $k\gg 0$, the Lefschetz fixed point formula expresses the trace of $γ_k$ in terms of fixed point data. More generally, one may consider the composition of $γ_k$ with the Toeplitz operator associated to some smooth function on $M$. Still more generally, in the presence of the compatible action of a compact and connected Lie group preserving $(M,L,γ)$, one may consider induce…
Let $M$ be complex projective manifold, and $A$ a positive line bundle on it. Assume that a compact and connected Lie group $G$ acts on $M$ in a Hamiltonian manner, and that this action linearizes to $A$. Then there is an associated unitary representation of $G$ on the associated algebro-geometric Hardy space. If the moment map is nowhere vanishing, the isotypical component are all finite dimensional, they are generally not spaces of sections of some power of $A$. One is then led to study the local and global asymptotic properties the isotypical component associated to a weight $k \, \boldsymb…
The main result of this paper is the description of asymptotics along rays in weight space of traces of equivariant Toeplitz operators composed with quantomorphisms for torus actions. The main ingredient in the proof is the microlocal analysis of the equivariant Szego kernels.
Let $X$ be the circle bundle associated to a positive line bundle on a complex projective (or, more generally, compact symplectic) manifold. The Tian-Zelditch expansion on $X$ may be seen as a local manifestation of the decomposition of the (generalized) Hardy space $H(X)$ into isotypes for the $S^1$-action. More generally, given a compatible action of a compact Lie group, and under general assumptions guaranteeing finite dimensionality of isotypes, we may look for asymptotic expansions locally reflecting the equivariant decomposition of $H(X)$ over the irreducible representations of the group…
We consider a natural variant of Berezin-Toeplitz quantization of compact Kähler manifolds, in the presence of a Hamiltonian circle action lifting to the quantizing line bundle. Assuming that the moment map is positive, we study the diagonal asymptotics of the associated Szegö and Toeplitz operators, and specifically their relation to the moment map and to the geometry of a certain symplectic quotient. When the underlying action is trivial and the moment map is taken to be identically equal to one, this scheme coincides with the usual Berezin-Toeplitz quantization. This continues previous work…
