Stable Maps into the Classifying Space of the General Linear Group


In this note we give a definition of stable maps into the classifying stack BGLr of the general linear group. To support our belief that the definition is the correct one, we show that there are natural boundary morphisms between the moduli groupoids parameterizing stable maps to BGLr. We warn the reader beforehand that we do not as yet have applications of our constructions. Our purpose in this note is mainly to explain the combinatorics of stable maps into BGLr. Before giving an idea of what stable maps to BGLr are, we first recall the case of stable maps into a complex projective variety V . Let g ∈ N0, and let S be a finite set. A stable S-pointed map of genus g consists of a prestable curve C together with nonsingular points (xs)s∈S on it and a morphism f : C → V of algebraic varieties such that the collection of these data has at most finitely many automorphisms. Stable maps into V are a fundamental notion for defining Gromov-Witten invariants for V . Namely, there is a Deligne-Mumford stack M g,S(V ) parameterizing all S-pointed stable maps of genus g, and Gromov-Witten invariants of V are defined by means of intersection theory on Mg,S(V ). The moduli stack M g,S(V ) breaks up into connected components M g,S(V, β) where β runs through some monoid. Stable maps which belong to that component are called of class β. Given two stable maps, say Si ∪ {∗i}-pointed, of genus gi and class βi for i = 1, 2 such that the points ∗i are mapped to the same point in V , one obtains a new stable map which is S1 ∪ S2-pointed, of genus g1 + g2 and class β1 + β2, by clutching together the two curves at the points ∗i. This yields a morphism

Cite this paper

@inproceedings{Kausz2006StableMI, title={Stable Maps into the Classifying Space of the General Linear Group}, author={I Kausz}, year={2006} }