Following the work of Mazzeo–Swoboda–Weiß–Witt [Duke Math. J. 165 (2016), 2227–2271] and Mochizuki [J. Topol. 9 (2016), 1021–1073], there is a map $\overline{\Xi }$ between the algebraic compactification of the Dolbeault moduli space of ${\rm SL}(2,\mathbb{C})$ Higgs bundles on a smooth projective curve coming from the $\mathbb{C}^\ast$ action and the analytic compactification of Hitchin’s moduli space of solutions to the $\mathsf{SU}(2)$ self-duality equations on a Riemann surface obtained by adding solutions to the decoupled equations, known as ‘limiting configurations’. This map extends the classical Kobayashi–Hitchin correspondence. The main result that this article will show is that $\overline{\Xi }$ fails to be continuous at the boundary over a certain subset of the discriminant locus of the Hitchin fibration.

We construct examples of compact homogeneous Riemannian manifolds admitting an invariant Bismut connection that is Ricci flat and non-flat, proving in this way that the generalized Alekseevsky–Kimelfeld theorem does not hold. The classification of compact homogeneous Bismut Ricci flat spaces in dimension $5$ is also provided. Moreover, we investigate compact homogeneous spaces with non-trivial third Betti number, and we point out other possible ways to construct Bismut Ricci flat manifolds. Finally, since Bismut Ricci flat connections correspond to fixed points of the generalized Ricci flow, we discuss the stability of some of our examples under the flow.

We study families of metrics on automorphic vector bundles associated with representations of the modular group. These metrics are defined using an Eisenstein series construction. We show that in certain cases, the residue of these Eisenstein metrics at their rightmost pole is a harmonic metric for the underlying representation of the modular group. The last section of the paper considers the case of a family of representations that are indecomposable but not irreducible. The analysis of the corresponding Eisenstein metrics, and the location of their rightmost pole, is an open question whose resolution depends on the asymptotics of matrix-valued Kloosterman sums.

We study a set $\mathcal{M}_{K,N}$ parameterising filtered SL(K)-Higgs bundles over $\mathbb{C}P^1$ with an irregular singularity at $z = \infty$, such that the eigenvalues of the Higgs field grow like $\vert \lambda \vert \sim \vert z^{N/K} \mathrm{d}z \vert$, where K and N are coprime. $\mathcal{M}_{K,N}$ carries a $\mathbb{C}^\times$-action analogous to the famous $\mathbb{C}^\times$-action introduced by Hitchin on the moduli spaces of Higgs bundles over compact curves. The construction of this $\mathbb{C}^\times$-action on $\mathcal{M}_{K,N}$ involves the rotation automorphism of the base $\mathbb{C}P^1$. We classify the fixed points of this $\mathbb{C}^\times$-action, and exhibit a curious 1-1 correspondence between these fixed points and certain representations of the vertex algebra $\mathcal{W}_K$ ; in particular we have the relation $\mu = {k-1-c_{\mathrm{eff}}}/{12}$ , where $\mu$ is a regulated version of the L2 norm of the Higgs field, and $c_{\mathrm{eff}}$ is the effective Virasoro central charge of the corresponding W-algebra representation. We also discuss a Białynicki–Birula-type decomposition of $\mathcal{M}_{K,N}$ , where the strata are labeled by isomorphism classes of the underlying filtered vector bundles.

Given a compact Kähler manifold X, it is shown that pairs of the form $(E,\, D)$ , where E is a trivial holomorphic vector bundle on X, and D is an integrable holomorphic connection on E, produce a neutral Tannakian category. The corresponding pro-algebraic affine group scheme is studied. In particular, it is shown that this pro-algebraic affine group scheme for a compact Riemann surface determines uniquely the isomorphism class of the Riemann surface.

This article provides an account of the functorial correspondence between irreducible singular $G$-monopoles on ${{S}^{1}}\,\times \,\sum $ and $\vec{t}$-stable meromorphic pairs on $\sum $. A theorem of B.Charbonneau and J. Hurtubise is thus generalized here from unitary to arbitrary compact, connected gauge groups. The required distinctions and similarities for unitary versus arbitrary gauge are clearly outlined, and many parallels are drawn for easy transition. Once the correspondence theorem is complete, the spectral decomposition is addressed.

We show that there is a bijective correspondence between the polystable parabolic principal $G$-bundles and solutions of the Hermitian-Einstein equation.

In this paper, we first investigate the Dirichlet problem for coupled vortex equations. Secondly, we give existence results for solutions of the coupled vortex equations on a class of complete noncompact Kähler manifolds which include simply-connected strictly negative curved manifolds, Hermitian symmetric spaces of noncompact type and strictly pseudo-convex domains equipped with the Bergmann metric.

This paper is one step toward infinite energy gauge theory and the geometry of infinite dimensional moduli spaces. We generalize a gluing construction in the usual Yang-Mills gauge theory to an “infinite energy” situation. We show that we can glue an infinite number of instantons, and that the resulting ASD connections have infinite energy in general. Moreover they have an infinite dimensional parameter space. Our construction is a generalization of Donaldson’s “alternating method”.

In this paper, we consider Hermitian harmonic maps from Hermitian manifolds into convex balls. We prove that there exist no non-trivial Hermitian harmonic maps from closed Hermitian manifolds into convex balls, and we use the heat flow method to solve the Dirichlet problem for Hermitian harmonic maps when the domain is a compact Hermitian manifold with non-empty boundary.

We introduce complex differential geometry twisted by a real line bundle. This provides a new approach to understand the various real objects that are associated with an anti-holomorphic involution. We also generalize a result of Greenleaf about real analytic sheaves from dimension 2 to higher dimensions.

The aim here is to define connections on a parabolic principal bundle. Some applications are given.

In this paper, first, we will investigate the Dirichlet problem for one type of vortex equation, which generalizes the well-known Hermitian Einstein equation. Secondly, we will give existence results for solutions of these vortex equations over various complete noncompact Kähler manifolds.

Let (X, ω) be a weakly pseudoconvex Kähler manifold, Y ⊂ X a closed submanifold defined by some holomorphic section of a vector bundle over X, and L a Hermitian line bundle satisfying certain positivity conditions. We prove that for any integer k > 0, any section of the jet sheaf which satisfies a certain L2 condition, can be extended into a global holomorphic section of L over X whose L2 growth on an arbitrary compact subset of X is under control. In particular, if Y is merely a point, this gives the existence of a global holomorphic function with an L2 norm under control and with prescribed values for all its derivatives up to order k at that point. This result generalizes the L2 extension theorems of Ohsawa-Takegoshi and of Manivel to the case of jets of sections of a line bundle. A technical difficulty is to achieve uniformity in the constant appearing in the final estimate. To this end, we make use of the exponential map and of a Rauch-type comparison theorem for complete Riemannian manifolds.

On a complex curve, we establish a correspondence between integrable connections with irregular singularities, and Higgs bundles such that the Higgs field is meromorphic with poles of any order. Moduli spaces of these objects are obtained with fixed generic polar parts at each singularity, which amounts to fixing a coadjoint orbit of the group $GL_r(\mathbb{C}[z]/z^n)$. We prove that they carry complete hyper-Kähler metrics.

Cet article de survol est le résumé de la conférence Coxeter-James de l'auteur, prononcée à la réunion d'hiver 1993 de la Société Mathématique du Canada.

La théorie de Morse décrit les liens entre la topologie d'une variété et la topologie des points critiques d'une fonction sur cette variété. La fonctionnelle d'énergie pour les applications d'une surface dans une variété, dont les points critiques seront des applications harmoniques et parfois holomorphes, et la fonctionnelle de Yang-Mills pour des connections sur une variété de dimension quatre sont deux cas en dimension infinie pour lesquels la théorie de Morse ne tient pas. Néanmois, dans les deux cas, on peut récupérer une quantité étonnante d'information, pourvu qu'on stabilise par rapport à un degré ou une charge qui sont des données du problème. Les preuves recyclent des résultats de la théorie de l'homotopie des années '70, et les combinent à des idées de géométrie complexe pour donner de jolis modèles des espaces en cause en termes de "particules". Nous espérons donner un survol général et accessible des idées utilisées.