Stack (mathematics)

Abstract: IN this paper, we present a framework for studying moduli spaces of finite dimensional representations of an arbitrary finite dimensional algebra A over an algebraically closed field k. (The abelian category of such representations is denoted by mod-A.) Our motivation is twofold. Firstly, such moduli spaces should play an important role in organising the representation theory of wild algebras. Secondly, such moduli spaces can be identified with moduli spaces of vector bundles on special projective varieties. This identification is somewhat hidden in earner work ([6], [7]) but has become more explicit recently ([4], [12]). It can now be seen to arise from a 'tilting equivalence' between the derived category of mod-A and the derived category of coherent sheaves on the variety. It is well-established that mod-A is equivalent to the category of representations of an arrow diagram, or 'quiver', Q by linear maps satisfying certain 'admissible' relations. Thus, the problem of classifying A -modules with a fixed class in the Grothendieck group K0(mod-A), represented by a 'dimension vector' a, is converted into one of classifying orbits for the action of a reductive algebraic group GL(a) on a subvariety VA(a) of the representation space 9t{Q, a) of the quiver. Now, the moduli spaces provided by classical invariant theory ([1], [18]) are not interesting in this context. This is because the classical theory only picks out the closed GL(a)-orbits in VA{a), which correspond to semisimple /4-modules, and the quiver Q is chosen so that there is only one semisimple A -module of each dimension vector. On the other hand, we can apply Mumford's geometric invariant theory, with the trivial linearisation twisted by a character x of GL(a), which restricts our attention to an open subset of VA(a), consisting of semistable representations. Within this open set there are more closed orbits and the corresponding algebraic quotient is then a more interesting moduli space. In fact, this approach also has a classical flavour, since it involves the relative (or semi-) invariants of the GL(a) action. The main purpose of this paper is to show that the notions of semistability and stability, that arise from the geometric invariant theory, coincide with more algebraic notions, expressed in the language of mod-A Indeed, the definition is formulated for an arbitrary abelian category as follows:

Abstract: Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmuller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces. It is related to the motivic dilogarithm.

Abstract: Parameter spaces: constructions and examples * Basic facts about moduli spaces of curves * Techniques * Construction of M_g * Limit Linear Series and the Brill-Noether Theory * Geometry of moduli spaces: Selected Results.