Related papers: Kirwan map and moduli space of flat connections

Let $G$ be a simple complex Lie group, $\alg{g}$ be its Lie algebra, $K$ be a maximal compact form of $G$ and $\alg{k}$ be a Lie algebra of $K$. We denote by $X\rightarrow \overline{X}$ the anti-involution of $\alg{g}$ which singles out the…

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…

Components of the Moduli space of sheaves on a K3 surface are parametrized by a lattice; the (algebraic) Mukai lattice. Isometries of the Mukai lattice often lift to symplectic birational isomorphisms of the collection of components. An…

There are two canonical projective structures on any compact Riemann surface of genus at least two: one coming from the uniformization theorem, and the other from Hodge theory. They produce two (different) families of projective structures…

We extend the notion of an almost flat bundle over a closed Riemannian manifold to bundles over simplicial complexes, and prove that up to a constant factor, this notion is invariant under pullback via maps which induce isomorphisms on…

Consider a Hamiltonian action of a compact connected Lie group $G$ on an aspherical symplectic manifold $(M,\omega)$. Under suitable assumptions, counting gauge equivalence classes of (symplectic) vortices on the plane $R^2$ conjecturally…

Let M be a moduli space of stable sheaves on a K3 or Abelian surface S. We express the class of the diagonal in the cartesian square of M in terms of the Chern classes of a universal sheaf. Consequently, we obtain generators of the…

We compute the intersection Betti numbers of the GIT model of the moduli space of numerically polarized Enriques surfaces of degree 2. The strategy of the cohomological calculation relies on a general method developed by Kirwan to compute…

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 $\kappa:H_G^*(M)\to H^*(M//G)$ is generated by a small number of classes $\alpha\in H_G^*(M)$…

We show that a flat principal bundle with compact connected structure group and its adjoint bundles of Lie groups have the same cohomology as the trivial bundle, which is done by proving they satisfy the condition for the Leray-Hirsch…

Let $G \to P \to M$ be a flat principal bundle over a closed and oriented manifold $M$ of dimension $m=2d$. We construct a map of Lie algebras $\Psi: \H_{2\ast} (L M) \to {\o}(\Mc)$, where $\H_{2\ast} (LM)$ is the even dimensional part of…

In this short note, we compute the Betti numbers of the moduli stack of flat SU(3)-bundles over a Klein bottle. We also handle the general compact group case over RP^2. In all cases the cohomology is found to be equivariantly formal,…

Let $M$ be a smooth manifold. We use Chern-Weil theory to study the characteristic classes of principal $G$-bundles built from continuous families of $\pi_{1}(M)$-representations, where $G$ is a compact Lie group. We then relate these…

We study the moduli space of Higgs bundles on a compact Riemann surface. It was shown by Thaddeus and Hausel (in rank 2) and Markman (in general rank) that the rational cohomology ring of this space is generated by universal classes. In…

Let $g$ be locally homogeneous (LH) Riemannian metric on a differentiable compact manifold $M$, and $K$ be a compact Lie group endowed with an $\mathrm {ad}$-invariant inner product on its Lie algebra $\mathfrak{k}$. A connection $A$ on a…

In [arXiv:2008.04625] the authors constructed a classifying space for polystable holomorphic vector bundles on a compact K\"ahler manifold using analytic GIT theory. The aim of this article is to show that this classifying space taken in…

We construct a universal partial compactification of the relative moduli space of semistable meromorphic Higgs bundles over the stack of stable pointed curves. It parametrizes meromorphic Gieseker Higgs bundles, and is equipped with a flat…

A Hamiltonian bundle $M \hookrightarrow P \to X$ (with monotone compact fibers) induces via Floer theory a type of ``bundle of $A _{\infty}$ categories'' over $X$, with fiber given by the Fukaya category of $M$. Morita theory of $A…

Given a compact, connected Lie group $K$, we use principal $K$-bundles to construct manifolds with prescribed finite-dimensional algebraic models. Conversely, let $M$ be a compact, connected, smooth manifold which supports an almost free…

We give a differential geometric construction of the holomorphic family of Higgs bundle moduli spaces over a curve C as a fibration over Teichm\"uller space. The method uses a function f defined on the character variety, essentially the…