Connectivity and Kirwan surjectivity for isoparametric submanifolds

Basic Kirwan Surjectivity for K-Contact Manifolds

We prove an analogue of Kirwan surjectivity in the setting of equivariant basic cohomology of K-contact manifolds. If the Reeb vector field induces a free $S^1$-action, the $S^1$-quotient is a symplectic manifold and our result reproduces Kirwan's surjectivity for these symplectic manifolds. We further prove a Tolman-Weitsman type description of the kernel of the basic Kirwan map for $S^1$-actions and show that torus actions on a K-contact manifold that preserve the contact form and admit 0 as a regular value of the contact moment map are equivariantly formal in the basic setting.

Morse Theory and Hyperkahler Kirwan Surjectivity for Higgs Bundles

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.

Kirwan Surjectivity for the Equivariant Dolbeault cohomology

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…

广义超凯勒约化的 Kirwan 满射性与 Lefschetz-Sommese 定理

Kirwan surjectivity and Lefschetz-Sommese theorems for a generalized hyperkähler reduction

研究了一类具有特定等变性质和（半）线性型 $G$-作用的 Hamiltonian $(G \ imes S^{1})$-流形，定义了超凯勒约化的类似物，并证明了 Kirwan 映射类似物的满射性，该结果适用于具有三哈密顿环面作用的超凯勒流形。

Surjectivity for Hamiltonian Loop Group Spacees

Let $G$ be a compact Lie group, and let $LG$ denote the corresponding loop group. Let $(X,ω)$ be a weakly symplectic Banach manifold. Consider a Hamiltonian action of $LG$ on $(X,ω)$, and assume that the moment map $μ: X \to L\fg^*$ is proper. We consider the function $|μ|^2: X \to \R$, and use a version of Morse theory to show that the inclusion map $j:μ^{-1}(0)\to X$ induces a surjection $j^*:H_G^*(X) \to H_G^*(μ^{-1}(0))$, in analogy with Kirwan's surjectivity theorem in the finite-dimensional case. We also prove a version of this surjectivity theorem for quasi-Hamiltonian $G$-spaces.

Morse theory of the moment map for representations of quivers

The results of this paper concern the Morse theory of the norm-square of the moment map on the space of representations of a quiver. We show that the gradient flow of this function converges, and that the Morse stratification induced by the gradient flow co-incides with the Harder-Narasimhan stratification from algebraic geometry. Moreover, the limit of the gradient flow is isomorphic to the graded object of the Harder-Narasimhan-Jordan-Hölder filtration associated to the initial conditions for the flow. With a view towards applications to Nakajima quiver varieties we construct explicit local …