FINE AND COARSE MODULI SPACES 3
This section overlaps with expository articles by Bongartz [5] and Geiss [13]. The
modest overlap is required for a consistent development of the subsequent ideas.
Then we return to Riemann’s classification program and discuss/exemplify the
concepts of a fine/coarse moduli space in the representation-theoretic context (Sec-
tion 4). To date, there are two different strategies to get mileage out of the con-
ceptual framework. In light of the fact that fine or coarse moduli spaces for the
full collection of isomorphism classes of Λ-modules with a given dimension vector
hardly ever exist, each method proposes a mode of slicing Λ-mod so as to extract
portions on which the conceptual vehicle of moduli spaces acquires traction. The
strategies of slicing take advantage of the particulars of the initial parametrizing
setups, and hence, in each case, specific methodology is called for to match the
target. Since there exist two prior survey articles dealing with Approach A, by
Geiss [13] and Reineke [29], we will give more room to Approach B in the present
overview.
One of the methods mimicks a strategy Mumford used in the classification of
vector bundles on certain projective varieties. It was adapted to the representation-
theoretic setting by King in [23] (see Section 5). Starting with an additive function
θ : K0(Λ) =
Zn
Z, King focuses on the Λ-modules with dimension vector d
which are θ-stable, resp. θ-semistable; interprets these stability conditions in terms
of the behavior of θ on submodule lattices; and shows how to apply techniques
from geometric invariant theory to secure a fine, resp. coarse, moduli space for θ-
(semi)stable modules. The resulting stability classes are not a priori representation-
theoretically distinguished, whence a fundamental challenge lies in “good” choices
of the function θ and a solid grasp of the corresponding θ-(semi)stable modules.
As this method is based on the affine parametrizing variety Modd(Λ), crucially
leaning on the features of this setup, it will be labeled Approach A. So far, its main
applications are to the hereditary case Λ = KQ, even though, in principle, King
extended the method to include arbitrary path algebras modulo relations.
By contrast, the second approach (labeled Approach B and described in Sec-
tions 6-8) starts with classes C of modules over Λ = KQ/I which are cut out by
purely representation-theoretic features, and aims at understanding these classes
through an analysis of the subvarieties of GRASSd(Λ) that encode them. The name
of the game is to exploit projectivity of the parametrizing variety and the typically
large unipotent radical of the acting group to find useful necessary and sufficient
conditions for the existence of a geometric quotient of the subvariety encoding C,
and to subsequently establish such a quotient as a moduli space that classifies
the representations in C up to isomorphism. Simultaneously, one seeks theoretical
and/or algorithmic access to moduli spaces whenever existence is guaranteed.
In describing either method, we state sample theorems witnessing viability
and illustrate them with examples. Each of the two outlines will conclude with a
discussion of pros and cons of the exhibited approach.
Acknowledgements
I wish to thank the organizers of the Auslander Conference and Distinguished
Lecture Series (Woods Hole, April 2012), K. Igusa, A. Martsinkovsky, and
G. Todorov, and the organizers F. Bleher and C. Chindris of the Conference on
Geometric Methods in Representation Theory (University of Missouri-Columbia,
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