3. Moduli spaces of p-divisible groups

International Journal of Mathematics, 2011

We provide a sketch of the GIT construction of the moduli spaces for the three classes of connections: the class of meromorphic connections with fixed divisor of poles D and its subclasses of integrable and integrable logarithmic connections. We use the Luna Slice Theorem to represent the germ of the moduli space as the quotient of the Kuranishi space by the automorphism group of the central fiber. This method is used to determine the singularities of the moduli space of connections in some examples.

2015

One of the main themes of this long article is the study of projective varieties which are K(H,1)'s, i.e. classifying spaces BH for some discrete group H. After recalling the basic properties of such classifying spaces, an important class of such varieties is introduced, the one of Bagnera-de Franchis varieties, the quotients of an Abelian variety by the free action of a cyclic group. Moduli spaces of Abelian varieties and of algebraic curves enter into the picture as examples of rational K(H,1)'s, through Teichmueller theory. The main thrust of the paper is to show how in the case of K(H,1)'s the study of moduli spaces and deformation classes can be achieved through by now classical results concerning regularity of classifying maps. The Inoue type varieties of Bauer and Catanese are introduced and studied as a key example, and new results are shown. Motivated from this study, the moduli spaces of algebraic varieties, and especially of algebraic curves with a group of au...

Journal of Differential Geometry, 1984

Analytic and Algebraic Geometry, 2017

We give a summary of joint work with Michael Thaddeus that realizes toroidal compactifcations of split reductive groups as moduli spaces of framed bundles on chains of rational curves. We include an extension of this work that covers Artin stacks with good moduli spaces. We discuss, for complex groups, the symplectic counterpart of these compactifications, and conclude with some open problems about the moduli problem concerned.

Topology, 1987

the moduli space of smooth h-pointed curves of genus g over C and by &is,h its natural compactification by means of stable curves. It is known that the Picard group of M,,, is a free Abelian group on h + 1 generators when g 2 3. This is due to Harer [4, 51 (cf. the Appendix). Instead of dealing with the Picard group of the moduli space it is usually more convenient, from a technical point of view, to work with the so-called Picard group of the moduli functor (see below for a precise definition), which we shall denote by Pit (JY~,,) if we are restricting to smooth curves and by Pit #&if we are allowing singular stable curves as well. As Mumford observes in [S], Pit (Jg.J has no torsion and contains Pit (MgJ as a subgroup of finite index (a proof of this will be sketched in the Appendix). The purpose of this note is to exhibit explicit bases for Pit (&& and for Pit (d&g,h)r which is also a free Abelian group. This is done in Theorem 2 (53), of which Theorem 1 in $2 is a special case. We shall now say a couple of words about our terminology. A family of h-pointed stable curves of genus g parametrized by S is a proper flat morphism II : V + S together with disjoint sections ol,. .. , a,, having the following properties. Each fiber n-'(s) is a connected curve of genus g having only nodes as singularities and such that each of its smooth rational components contains at least three points belonging to the union of the remaining components and of the sections; moreover, for each i, Go is a smooth point of n-'(s). Following Mumford [7,8], by a line bundle on the moduli functor Jjtg,h we mean the datum of a line bundle L, (often written L,) on S for any family F = (n : %Z + S, (rl,. .. , CT,,) of h-pointed stable curves of genus g, and of an isomorphism L, s cz*(L,) for any Cartesian square of families of h-pointed stable curves; these isomorphisms are moreover required to satisfy an obvious cocycle condition. It is important to notice that we get an equivalent definition if, in the above, we restrict to families of pointed stable curves which are, near any point of the base, universal deformations for the corresponding fiber. We write Pit (Sg,h) to denote the group l Supported in part by grants from the C.N.R. and the Italian Ministry of Public Education.

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

This text aims to explain what topology, at present, has to say about a few of the many moduli spaces that are currently under study in mathematics. The most prominent one is the moduli space M g of all Riemann surfaces of genus g. Other examples include the Gromov-Witten moduli space of pseudo-holomorphic curves in a symplectic background, the moduli space of graphs and Waldhausen's algebraic K-theory of spaces.

Mathematische Nachrichten, 2004

Japanese journal of mathematics. New series, 1992

In this paper we introduce a new invariant for the action of a finite group $G$ on a compact complex curve of genus $g$. With the aid of this invariant we achieve the classification of the components of the locus (in the moduli space) of curves admitting an effective action by the dihedral group $D_n$. This invariant could be useful in order to extend the results of Livingston and Dunfield and Thurston to the ramified case.

Let M and N be even-dimensional oriented real manifolds, and $u:M \to N$ be a smooth mapping. A pair of complex structures at M and N is called u-compatible if the mapping u is holomorphic with respect to these structures. The quotient of the space of u-compatible pairs of complex structures by the group of u-equivariant pairs of diffeomorphisms of M and N is called a moduli space of u-equivariant complex structures. The paper contains a description of the fundamental group G of this moduli space in the following case: $N = CP^1, M \subset CP^2$ is a hyperelliptic genus g curve given by the equation $y^2 = Q(x)$ where Q is a generic polynomial of degree 2g+1, and $u(x,y) = y^2$. The group G is a kernel of several (equivalent) actions of the braid-cyclic group $BC_{2g}$ on 2g strands. These are: an action on the set of trees with 2g numbered edges, an action on the set of all splittings of a (4g+2)-gon into numbered nonintersecting quadrangles, and an action on a certain set of subgr...

We calculate the moduli space of flat SU(2) connections on several surfaces. The surfaces in question are the torus with zero and one puncture, and the sphere with one, two, three and four punctures. Included is also a chapter with serveral nice facts about SU(2) and its lie algebra. Besides all this is a chapter dedicated to the SU(2) representation variety, discussing its topology and the dimension in the case of a surface group.

Journal of Algebra, 1991

Journal of Pure and Applied Algebra, 1998

We show that any finite mixed Lie algebra in the sense of Lazard, of characteristic p and length 3, is the associated graded of some finite p-filtered group. This amounts to calculating the second mod p cohomology group of a p-group of p-class 2 in terms of the mixed Lie algebra associated to its Frattini filtration.

Bulletin des Sciences Mathématiques, 2011

Let G be a connected complex semisimple affine algebraic group, and let K be a maximal compact subgroup of G. Let X be a noncompact oriented surface. The main theorem of Florentino and Lawton (2009) [3] says that the moduli space of flat K-connections on X is a strong deformation retraction of the moduli space of flat G-connections on X. We prove that this statement fails whenever X is compact of genus at least two.

2007

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