A BRIEF SURVEY OF HIGGS BUNDLES

Considering a compact Riemann surface of genus greater or equal than two, a Higgs bundle is a pair composed of a holomorphic bundle over the Riemann surface, joint with an auxiliar vector field, so-called Higgs field. This theory started around thirty years ago, with Hitchin’s work, when he reduced the self-duality equations from dimension four to dimension two, and so, studied those equations over Riemann surfaces. Hitchin baptized those fields as Higgs fields because in the context of physics and gauge theory, they describe similar particles to those described by the Higgs bosson. Later, Simpson used the name Higgs bundle for a holomorphic bundle together with a Higgs field. Today, Higgs bundles are the subject of research in several areas such as non-abelian Hodge theory, Langlands, mirror symmetry, integrable systems, quantum field theory (QFT), among others. The main purposes here are to introduce these objects, and to present a brief but complete construction of the moduli space of Higgs bundles.

Multiplicity-freeness of Unitary Representations in Sections of Holomorphic Hilbert Bundles

We prove several results asserting that the action of a Banach–Lie group on Hilbert spaces of holomorphic sections of a holomorphic Hilbert space bundle over a complex Banach manifold is multiplicity-free. These results require the existence of compatible anti-holomorphic bundle maps and certain multiplicity-freeness assumptions for stabilizer groups. For the group action on the base, the notion of an $(S,\sigma )$-weakly visible action (generalizing T. Koboyashi’s visible actions) provides an effective way to express the assumptions in an economical fashion. In particular, we derive a version for group actions on homogeneous bundles for larger groups. We illustrate these general results by several examples related to operator groups and von Neumann algebras.

HODGE POLYNOMIALS OF THE MODULI SPACES OF PAIRS

Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic pair on X is a couple (E, ϕ), where E is a holomorphic bundle over X of rank n and degree d, and ϕ ∈ H0(E) is a holomorphic section. In this paper, we determine the Hodge polynomials of the moduli spaces of rank 2 pairs, using the theory of mixed Hodge structures. We also deal with the case in which E has fixed determinant.

On the extension of L2 holomorphic functions V-Effects of generalization

A general extension theorem for L2 holomorphic bundle-valued top forms is formulated. Although its proof is based on a principle similar to Ohsawa-Takegoshi’s extension theorem, it explains previous L2 extendability results systematically and bridges extension theory and division theory.

ANALYTIC CONTINUATION OF VECTOR BUNDLES WITH Lp –CURVATURE

This article addresses the problem of removable singularities for a Hermitian-holomorphic vector bundle ℰ, defined on the complement of an analytic set A of complex codimension at least two in a complex n-dimensional manifold X. In particular it is shown here that there exists a unique holomorphic bundle [Formula: see text] on X, such that [Formula: see text], when the curvature of ℰ belongs to Ln (X\A). This result is in fact sharp, as counterexamples exist for the extensibility of ℰ with curvature in Lp, p < n. Extension across general closed subsets of finite (2n - 4)-dimensional Hausdorff measure then follows directly from a slicing theorem of Bando and Siu.