Morse theory of the moment map for representations of quivers

通过 Morse 复形上的杯积实现箭图簇的卷积

Convolution for quiver varieties via cup product on a Morse complex

本文通过 Morse 复形上的杯积编码拓扑信息，为 Nakajima 箭图簇的卷积表示构造提供新方法，并发展了弱横截性理论以处理矩映射范数平方情形。

Moment map flows and the Hecke correspondence for quivers

In this paper we investigate the convergence properties of the upwards gradient flow of the norm-square of a moment map on the space of representations of a quiver. The first main result gives a necessary and sufficient algebraic criterion for a complex group orbit to intersect the unstable set of a given critical point. Therefore we can classify all of the isomorphism classes which contain an initial condition that flows up to a given critical point. As an application, we then show that Nakajima's Hecke correspondence for quivers has a Morse-theoretic interpretation as pairs of critical point…

The quiver at the bottom of the twisted nilpotent cone on $\mathbb P^1$

For the moduli space of Higgs bundles on a Riemann surface of positive genus, critical points of the natural Morse-Bott function lie along the nilpotent cone of the Hitchin fibration and are representations of $\mbox{A}$-type quivers in a twisted category of holomorphic bundles. The critical points that globally minimize the function are representations of $\mbox{A}_1$. For twisted Higgs bundles on the projective line, the quiver describing the bottom of the cone is more complicated. We determine it here. We show that the moduli space is topologically connected whenever the rank and degree are…

Morse theory on Hamiltonian G-spaces and equivariant K-theory

Let $G$ be a torus and $M$ a compact Hamiltonian $G$-manifold with finite fixed point set $M^G$. If $T$ is a circle subgroup of $G$ with $M^G=M^T$, the $T$-moment map is a Morse function. We will show that the associated Morse stratification of $M$ by unstable manifolds gives one a canonical basis of $K_G(M)$. A key ingredient in our proof is the notion of local index $I_p(a)$ for $a\in K_G(M)$ and $p\in M^G$. We will show that corresponding to this stratification there is a basis $τ_p$, $p\in M^G$, for $K_G(M)$ as a module over $K_G(\pt)$ characterized by the property: $I_qτ_p=δ^q_p$. For $M$…

Trivializations of moment maps

We study various trivializations of moment maps. First in the general framework of a reductive group $G$ acting on a smooth affine variety. We prove that the moment map is a locally trivial fibration over a regular locus of the center of the Lie algebra of $H$ a maximal compact subgroup of $G$. The construction relies on Kempf-Ness theory and Morse theory of the square norm of the moment map studied by Kirwan, Ness-Mumford and Sjamaar. Then we apply it together with ideas from Nakajima and Kronheimer to trivialize the hyperkaehler moment map for Nakajima quiver varieties. Notice this trivializ…

辛映射与超Kähler动量映射几何

Symplectic maps and hyperKähler moment map geometry

本文建立了4维实环面辛微分同胚群与超Kähler动量映射零点轨迹的对应关系，提出修正动量映射流，推广至多面体设定并证明Duistermaat型定理，为多面体辛几何与Morse-Bott理论奠定基础。