This article has been merged with arXiv:1106.1729. The new article is: A Quantum Kirwan Map: Bubbling and Fredholm Theory for Symplectic Vortices over the Plane, arXiv:1209.5866
Consider a Hamiltonian action of a compact connected Lie group $G$ on an aspherical symplectic manifold $(M,ω)$. Under some assumptions on $(M,ω)$ and the action, D. A. Salamon conjectured that counting gauge equivalence classes of symplectic vortices on the plane $R^2$ gives rise to a quantum deformation $Qκ_G$ of the Kirwan map. This article is the first of three, whose goal is to define $Qκ_G$ rigorously. Its main result is that the vertical differential of the vortex equations over $R^2$ (at the level of gauge equivalence) is a Fredholm operator of a specified index. Potentially, the map $Qκ_G$ can be used to compute the quantum cohomology of many symplectic quotients. Conjecturally it also gives rise to quantum generalizations of non-abelian localization and abelianization (see [Woodward-Ziltener]).
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.…
Biinvariant diagonal classes give rise to right inverses of the Kirwan map. By means of multivalued perturbations of the gradient flow equation such classes are constructed explicitly for $S^1$-Hamiltonian spaces. Moreover, the notion of biinvariant diagonal class is extended to a cohomology map that gives extra topological information on the Hamiltonian space. The non-uniqueness of global biinvariant diagonal classes is also proven.
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 a Hamiltonian action of a compact connected Lie group $G$ on an aspherical symplectic manifold $(M,ω)$. Under suitable assumptions, counting gauge equivalence classes of (symplectic) vortices on the plane $R^2$ conjecturally gives rise to a quantum deformation $Qk_G$ of the Kirwan map. This is the second of a series of articles, whose goal is to define $Qk_G$ rigorously. The main result is that every sequence of vortices with uniformly bounded energies has a subsequence that converges to a genus 0 stable map of vortices on $R^2$ and holomorphic spheres in the symplectic quotient. Pote…
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], [J…
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 …
We study the symplectic vortex equation over the complex plane, for the target space ${\mathbb C}^N$ ($N\geq 2$) with diagonal U(1)-action. We classify all solutions with finite energy and identify their moduli spaces, which generalizes Taubes' result for N=1. We also studied their compactifications and use them to compute the associated quantum Kirwan maps $κ_Q: H^*_{U(1)}({\mathbb C}^N) \to QH^*({\mathbb P}^{N-1})$.
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…
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…
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…
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 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…
This is the first in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology $QH_G(X)$ of a smooth complex projective variety X with the action of a connected complex reductive group $G$ to the orbifold quantum cohomology $QH(X//G)$ of its geometric invariant theory quotient $X//G$, and prove that it intertwines the genus zero gauged Gromov-Witten potential of X with the genus zero Gromov-Witten graph potential of $X//G$.
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…
Given a Hamiltonian system $ (M,ω, G,μ) $ where $(M,ω)$ is a symplectic manifold, $G$ is a compact connected Lie group acting on $(M,ω)$ with moment map $ μ:M \rightarrow\mathfrak{g}^{*}$, then one may construct the symplectic quotient $(M//G, ω_{red})$ where $M//G := μ^{-1}(0)/G$. Kirwan used the norm-square of the moment map, $|μ|^2$, as a G-equivariant Morse function on $M$ to derive formulas for the rational Betti numbers of $M//G$. A real Hamiltonian system $(M,ω, G,μ, σ, ϕ) $ is a Hamiltonian system along with a pair of involutions $(σ:M \rightarrow M, ϕ:G \rightarrow G) $ satisfying cer…
We extend the famous convexity theorem of Atiyah, Guillemin and Sternberg to the case of non-Hamiltonian actions. We show that the image of a generalized momentum map is a bounded polytope times a vector space. We prove that this picture is stable for small perturbations of the symplectic form. We also observe that an n-dimensional torus symplectic action on a 2n-dimensional symplectic manifold, with fixed point, is Hamiltonian. We finally prove that an extension of Kirwan's convexity theorem (for compact group actions) is also true.
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…
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 $IH^*(X)$…
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…
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…
We prove a Hitchin-Kobayashi correspondence for affine vortices generalizing a result of Jaffe-Taubes for the action of the circle on the affine line. Namely, suppose a compact Lie group K has a Hamiltonian action on a Kaehler manifold X which is either compact or convex at infinity with a proper moment map, and so that stable=semistable for the action of the complexified Lie group G. Then, for some sufficiently divisible integer n, there is a bijection between gauge equivalence classes of K-vortices with target X modulo gauge and isomorphism classes of maps from the weighted projective line P…
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, $…
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 $i^*$.
By coupling a Hamiltonian mechanical system with a linear Hamiltonian field theory one obtains an infinite-dimensional Hamiltonian system with regularizing nonlinearity, where the underlying phase space is given by the product of a finite-dimensional symplectic manifold with an infinite-dimensional linear symplectic Hilbert space. After our compactness results we continue our program for defining a Floer homology theory for this class of infinite-dimensional Hamiltonian systems. Based on the presence of small divisors we introduce a new notion of nondegeneracy for time-periodic solutions which…
In this note we consider a quantum reduction scheme in deformation quantization on symplectic manifolds proposed by Bordemann, Herbig and Waldmann based on BRST cohomology. We explicitly construct the induced map on equivalence classes of star products which will turn out to be an analogue to the Kirwan map but in the Cartan model of equivariant cohomology. As a byproduct we shall see that every star product on a (suitable) reduced manifold is equivalent to a reduced star product.
Assume $(X, ω)$ 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 $μ$. 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 $μ^{-1}(0)$ 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 morp…
