Moduli spaces of holomorphic triples over compact Riemann surfaces

Abstract

A holomorphic triple over a compact Riemann surface consists of two holomorphic vector bundles and a holomorphic map between them. After fixing the topological types of the bundles and a real parameter, there exist moduli spaces of stable holomorphic triples. In this paper we study non-emptiness, irreducibility, smoothness, and birational descriptions of these moduli spaces for a certain range of the parameter. Our results have important applications to the study of the moduli space of representations of the fundamental group of the surface into unitary Lie groups of indefinite signature, which we explore in a companion paper "Surface group representations in PU(p,q) and Higgs bundles". Another application, that we study in this paper, is to the existence of stable bundles on the product of the surface by the complex projective line. This paper, and its companion mentioned above, form a substantially revised version of math.AG/0206012.

44 pages. v2: minor clarifications and corrections, to appear in Math. Annalen

Cited by in corpus (19)

• Moduli of ADHM Sheaves and Local Donaldson-Thomas Theory
• Homotopy groups of moduli spaces of representations
• On the geometry of moduli spaces of holomorphic chains over compact Riemann surfaces
• Chamber Structure and Wallcrossing in The ADHM Theory of Curves I
• Torelli theorem for the moduli spaces of pairs
• Aspects of the topology and combinatorics of Higgs bundle moduli spaces
• Introduction to Nonabelian Hodge Theory: flat connections, Higgs bundles and complex variations of Hodge structure
• Rank two quadratic pairs and surface group representations
• Fourier-Mukai and Nahm transforms for holomorphic triples on elliptic curves
• Irreducibility of moduli of semistable Chains and applications to U(p,q)-Higgs bundles
• Twisted argyle quivers and Higgs bundles
• The quiver at the bottom of the twisted nilpotent cone on ${\mathbb{P}}^1$
• Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces
• Birationality of moduli spaces of twisted $\mathrm{U}(p,q)$-Higgs bundles
• Simpson Filtration and Oper Stratum Conjecture
• Quiver Bundles and Wall Crossing for Chains
• Tensor Products of Quiver Bundles
• Kähler-Yang-Mills Equations and Vortices
• Non-abelian vortices on CP^1 and Grassmannians