Georgios Daskalopoulos, Jonathan Weitsman, Richard Wentworth, Graeme Wilkin
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.
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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…
Integral Kirwan Surjectivity
本文改进 Hamiltonian G-作用的 Kirwan 满射性与形式性定理，证明稳定子阶取逆后的加法分裂并推广至 MU-模谱。
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 …
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。
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 …
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…
Lagrangian Intersections, Symplectic Reduction and Kirwan Surjectivity
本文研究了辛约简与基万覆盖性，通过拉格朗日交集和共轭序列分裂，推导了辛商空间中的同调关系，并探讨了3d B模型中的对称性与约简问题的关联。
This paper gives a partial desingularisation construction for hyperkähler quotients and a criterion for the surjectivity of an analogue of the Kirwan map to the cohomology of hyperkähler quotients. This criterion is applied to some linear actions on hyperkähler vector spaces.
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.
Equivariant Morse theory for Lie algebra actions on Riemannian foliations
该研究探讨黎曼叶状结构上李代数作用的等变莫尔斯理论，建立相关引理与定理，推广基尔万定理并给出黎曼叶状结构莫尔斯不等式的新证明。
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…
Singular cohomology of symplectic quotients by circle actions and Kirwan surjectivity
本文研究圆周 Hamiltonian 作用辛商的奇异上同调，通过部分消奇异化上的消解微分形式复形和长正合列，证明偶数度 Kirwan 满射性，并以阿贝尔多边形空间为例。
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…
Here we prove the necessary analytic results to construct a Morse theory for the Yang-Mills-Higgs functional on the space of Higgs bundles over a compact Riemann surface. The main result is that the gradient flow with initial conditions $(A'', ϕ)$ converges to a critical point of this functional, the isomorphism class of which is given by the graded object associated to the Harder-Narasimhan-Seshadri filtration of $(A'', ϕ)$. In particular, the results of this paper show that the failure of hyperkähler Kirwan surjectivity for rank 2 fixed determinant Higgs bundles does not occur because of a f…
We study the Morse theory of the Yang-Mills-Higgs functional on the space of pairs $(A,Φ)$, where $A$ is a unitary connection on a rank 2 hermitian vector bundle over a compact Riemann surface, and $Φ$ is a holomorphic section of $(E, d_A")$. We prove that a certain explicitly defined substratification of the Morse stratification is perfect in the sense of $\G$-equivariant cohomology, where $\G$ denotes the unitary gauge group. As a consequence, Kirwan surjectivity holds for pairs. It also follows that the twist embedding into higher degree induces a surjection on equivariant cohomology. This …
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