Some results on the moduli spaces of quiver bundles

Abstract

This article is concerned with the study of gauge theory, stability and moduli for twisted quiver bundles in algebraic geometry. We review natural vortex equations for twisted quiver bundles and their link with a stability condition. Then we provide a brief overview of their relevance to other geometric problems and explain how quiver bundles can be viewed as sheaves of modules over a sheaf of associative algebras and why this view point is useful, e.g., in their deformation theory. Next we explain the main steps of an algebro-geometric construction of their moduli spaces. Finally, we focus on the special case of holomorphic chains over Riemann surfaces, providing some basic links with quiver representation theory. Combined with the analysis of the homological algebra of quiver sheaves and modules, these links provide a criterion for smoothness of the moduli spaces and tools to study their variation with respect to stability.Keywords Quiver bundle · Vortex equations · Moduli space · Stability · Hitchin-Kobayashi correspondence Mathematics Subject Classification (2000) 14D20 · 16G20 · 14F05 · 32L05 · 53C07 This article is a review of several results about quiver bundles in algebraic and complex geometry, with emphasis on their gauge-theoretic equations, the algebro-geometric construction of their moduli spaces and some applications.A quiver, or directed graph, is a pair of sets Q = (Q 0 , Q 1 ) together with two maps h, t : Q 1 → Q 0 . The elements of Q 0 and Q 1 are respectively called the vertices and the arrows of the quiver. For each arrow a ∈ Q 1 , the vertices v = ta and w = ha are respectively called the tail and the head of the arrow a, which is denoted by a : v → w. We will always assume that the quiver is finite, i.e. the sets Q 0 and Q 1 are finite.Quivers provide concrete descriptions for many interesting categories in terms of their representations. A representation (E, φ) of Q in a category A is the data given by an L. Álvarez-Cónsul (B) Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Serrano 113 bis,

Balanced metrics on twisted Higgs bundles

“…A Hitchin-Kobayashi correspondence for these objects was proved in [2], relating the existence of solutions of the twisted quiver vortex equations with the slope stability of a twisted quiver bundle. A notion of Gieseker stability for twisted quiver sheaves has been provided in [1,34] for the construction of a moduli space. The twisted Higgs bundles we have considered above are precisely twisted Quiver bundles for the quiver consisting of a single vertex and arrow (with head and tail being this one vertex).…”

Stratifications of parameter spaces for complexes by cohomology types

“…The final step is give by choosing a linearisation on B σ and pulling this back to the parameter scheme T via 1). The linearisation on B σ is given by taking a weighted tensor product of these linearisations and twisting by a character ρ of G = G σ .…”

“…We consider the moduli of (isomorphism classes of) complexes of sheaves on X, or equivalently moduli of Q-sheaves over X where Q is the quiver • → • → · · · · · · → • → • with relations imposed to ensure the boundary maps square to zero. Moduli of quiver sheaves have been studied in [1,2,4,12]. There is a construction of moduli spaces of S-equivalence classes of 'semistable' complexes due to Schmitt [12] as a geometric invariant theory quotient of a reductive group G acting on a parameter space T for complexes with fixed invariants.…”

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

“…As will be reviewed in §2, a critical point must have a particular form, called a holomorphic chain. Their appearances in this context the literature include [13,4,1,26,32]. The Morse index can be read off directly from the chain.…”

“…In representation-theoretic terms, a holomorphic chain is a representation of an A-type quiver Q, but the representations are taken in a category of holomorphic bundles on X with twisted morphisms rather than the category of vector spaces. These objects are a special case of the "quiver bundles" considered in [14,1,21,29]. Here, we consider quivers Q with finite underlying graph A n for some n ≥ 1.…”