Topology of moduli spaces of free group representations in real reductive groups. An unexpected byproduct of this discovery was the realization that the analysis of more complex curve singularities is not a necessary step in understanding the geometry of the moduli. The most prominent one is the moduli space mg of all riemann surfaces of genus g. Permission is granted to copy, distribute andor modify this document under the terms of the gnu free documentation license, version 1. Some notes containing a brief survey of the cohomology of the moduli space of curves and the harerzagier formula for the orbifold euler characteristic. Topological moduli spaces of knots allen hatcher classical knot theory is concerned with isotopy classes of knots in the 3 sphere, in other words, pathcomponents of the space kof all smooth submanifolds of s3 diffeomorphic to the circle s1. Using wallcrossing for k3 surfaces, we establish birational equivalence of moduli spaces of stable objects on generic enriques surfaces for different stability conditions. Mumford, was that curves with only mild singularities form smooth compact moduli spaces. The generating function is expressed in terms of euler characteristics of certain moduli spaces of stable configurations of linear subspaces appearing in classical git. M theory moduli spaces and torsionfree structures mariana grana, c. Moduli spaces from a topological viewpoint ib madsen abstract.
The purpose of these lecture notes is to introduce the basics of the birational geometry of moduli spaces to students who have taken an introductory course in algebraic. In this paper we relate two important but disparate topics in the. The study of moduli spaces is an old branch of algebraic geometry with. There are further technical conditions that are omitted for the sake of brevity. Moduli spaces of sheaves of rank 2 on a surface are not smooth, unless we consider sheaves with special invariants on special surfaces. An introduction to the topology of the moduli space of stable. The construction of moduli spaces and geometric invariant theory by dinamo djounvouna in algebraic geometry, classi. The starting point is the remark that for any isomorphism. Topological moduli spaces of knots cornell university. This is also unfinished, but the aim is to describe the homotopy types of the components of the space of all knots in the 3sphere.
Combinatorial equivalence of real moduli spaces satyan l. There is also related work on the stabilization of the cohomology of the moduli spaces of locally free stable sheaves and the atiyahjones conjecture in the gauge theory literature see lij97, tau84, tau89. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. Moreover, we assume that p 1 and p 2 are dual in the sense of abelian varieties. This volume covers topics in the research area of birational geometry and moduli spaces, including irreducible holomorphic symplectic manifolds, severi varieties, degeneration of calabiyau varieties, toric fano threefolds, mirror symmetry, canonical bundle formula, the lefschetz principle and more. Clay mathematics institute workshop moduli spaces of vector bundles, with a view towards coherent sheaves october 611, 2006 cambridge, massachusetts clay mathematics proceedings volume 14 grassmannians, moduli spaces and vector bundles clay mathematics institute, cambridge, ma p. The argument makes use of a new chowtheoretic result, showing that moduli spaces on an enriques surface. People tend to say classifying space when in the context of topology, and they tend to say moduli space when in a context of complex geometry or algebraic geometry. In mik06 and mik07 mikhalkin investigated the moduli space of tropical curves of genus 0 with nmarked points and constructed a compacti cation which is the tropical analogue of the delignemumford compacti cation in algebraic geometry. An example of a different kind is the jacobian, which is a moduli space that classi. Moduli of graphs and automorphisms of free groups 93 it is also shown in 2 that the subspace of ip consisting of all length functions which arise from isometric actions of a given finitely generated group on rtrees is compact, as is the smaller space of length functions associated to. Moduli spaces of stable quotients and wallcrossing phenomena yukinobu toda abstract the moduli space of holomorphic maps from riemann surfaces to the grassmannian is known to have two kinds of compacti. Moduli spaces of special lagrangians and kahlereinstein. Geometry of moduli spaces and representation theory.
I added a few words outlining other work of harer on the stability of the cohomology of the moduli space of curves. In some special cases of moduli spaces of stable pairs, extii,e 0 only when i 0,1. Yoshioka computes the betti numbers of moduli spaces of rank 2 sheaves on p2 and proves the. Lecture notes mit opencourseware free online course. The book begins by studying individual smooth algebraic curves, including the most beautiful ones, before addressing families of curves. Moduli spaces of stable vector bundles on enriques surfaces. In these cases, we will demonstrate the existence of the virtual fundamental class, which is important for the study of intersection theory on the moduli space.
Download book an introduction to riemann surfaces algebraic curves and moduli spaces in pdf format. One also distinguishes between fine and coarse moduli spaces for the same moduli problem. Moduli spaces london mathematical society lecture note. A p 2, which act faithfully and transitively on the bres of m i. A copy of the license is included in the section entitled gnu free documentation license.
Further we show that the moduli space of these pairs is locally embedded in a certain relative cohomology group. In the example of circles, for instance, the moduli space is not just an abstract set, but the absolute value of the difference of the radii defines a metric for determining when two circles are close. Moduli spaces can be thought of as giving a universal space of parameters for the problem. An introduction to the topology of the moduli space of. Dynamics of teichmuller modular groups and topology of moduli. 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 basic problem is that of moduli of smooth complete curves of a fixed genus. A moduli space for finitely generated torsion free rmodules of rank 1 is constructed by g. Mcmullen proved that the moduli spaces of riemann surfaces are k. Notes on the construction of the moduli space of curves dan edidin the purpose of these notes is to discuss the problem of moduli for curves of genus g. Moduli theory is the study of how objects, typically in algebraic geometry but sometimes in other areas of mathematics, vary in families and is fundamental to an understanding of the objects themselves. Msri series volume 28, 1995 torelli groups and geometry of moduli spaces of curves richard m. Using the simplest methods, and proving as much as possible, formulas are derived for the betti numbers. Moduli problems and geometric invariant theory victoria hoskins abstract in this course, we study moduli problems in algebraic geometry and the construction of moduli spaces using geometric invariant theory.
We prove iqbals conjecture on the relationship between the free energy of closed string theory in local toric geometry and the wesszuminowitten model. The aim of this paper is to show how facts about the homology of the torelli group imply. Suppose that we want to classify some kind of object, for example. Architecture form, space, & order third edition architecture form, space, & order third edition. We also compare these real moduli spaces to the real points of the corresponding complex. Here different solutions are identified if they are isomorphic that is, geometrically the same. Abstract references similar articles additional information. Good geometry of the moduli spaces of riemann surfaces. We show that the image of the moduli space of stable bundles on an enriques surface by the pull back map is a lagrangian subvariety in the moduli space of stable bundles, which is a symplectic variety, on the covering k3 surface. Introduction let x be an artin stack always assumed to have quasicompact and separated diagonal over specz. An introduction to moduli spaces of curves and its. Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces. The torelli group tg is the group of isotopy classes of di eo morphisms of a compact orientable surface of genus g that act trivially on the homology of the surface.
Minicourse on moduli spaces emily clader june 2011 1 what is a moduli space. The rough idea is but see the caveat below that the term moduli space is essentially a synonym for representing object and for classifying space. The moduli space is, therefore, the positive real numbers. Compact moduli spaces of surfaces and exceptional vector bundles. Birational geometry and moduli spaces elisabetta colombo. This divisor class varies naturally with the bridgeland stability condition. This book is based on lectures given at the graduate summer school of the 2015 park city mathematics institute program geometry of moduli spaces and representation theory, and is devoted to several interrelated topics in algebraic geometry, topology of algebraic varieties, and representation theory. So the term fine moduli space mainly indicates that a given object that might be a delignemumford stack is actually just a plain scheme. Torelli groups and geometry of moduli spaces of curves. The boardman vogt resolution and tropical moduli spaces. You can read online an introduction to riemann surfaces algebraic curves and moduli spaces here in pdf, epub, mobi or docx formats. Moduli definition of moduli by the free dictionary. The lecture notes section lists the lecture topics, along with the lecture notes files, covered in the course.
This thesis is concerned with the application of operadic methods, particularly modular operads, to questions arising in the study of moduli spaces of surfaces as well as applications to the study of homotopy algebras and new constructions of quantum invariants of manifolds inspired by ideas originating from physics. We also describe singularities and some other features of it. Minicourse on moduli spaces university of michigan. For instance, if g is a linearly reductive group scheme acting linearly on x. Find materials for this course in the pages linked along the left. Compact moduli spaces of surfaces and exceptional vector bundles the moduli space of surfaces of general type has a natural compacti cation due to koll ar and shepherdbarron ksb88 which is analogous to the delignemumford compacti cation of the moduli space of curves dm69. When studying geometric objects, it is desirable to classify them according to different criteria in order to be able to distinguish the equivalent classes in this category. Torelli groups and geometry of moduli spaces of curves richard m.
Devadoss introduction the riemann moduli space mn gof surfaces of genus gwith nmarked points has become a central object in mathematical physics. One can consider moduli spaces of vector bundles, coherent sheaves, etc. Underthe same assumptions, the moduli spaces turn out to be irreducible. Topology of moduli space of representations sean lawton george mason university the geometry, topology and physics of moduli spaces of higgs bundles institute for mathematical sciences, national university of singapore july 22, 2014 lawton gmu topology of moduli spaces imsnus, july 2014. Euler characteristics of moduli spaces of torsion free.
Localizations on moduli spaces and free field realizations of feynman rules jian zhou abstract. Introduction hilbert schemes of points moduli of sheaves donaldson invariants curve counting generating functions of invariants of moduli spaces generating functions of invariants of moduli spaces moduli spaces mn depending on n 0. The torelli group tg is the group of isotopy classes of di eomorphisms of a compact orientable surface of genus g that act trivially on the homology of the surface. Pdf mtheory moduli spaces and torsionfree structures. Good moduli spaces appear to be the correct notion characterizing morphisms from stacks arising from quotients by linearly reductive groups to the quotient scheme. Let x be a smooth, complex projective surface, h a polarization on x. For n5, the classi cation of prehomogeneous vector spaces rules out such a simple interpretation, but it may still be possible to identify the moduli spaces as quotients of other simple varieties by group actions.
Curves of genus g, onedimensional subspaces of rn, finite sets. Moduli spaces are spaces of solutions of geometric classification problems. This textbook offers a concise yet thorough introduction to the notion of moduli spaces of complex algebraic curves. A p 2, which act faithfully and transitively on the bres of. Its importance was emphasized by grothendieck in his famous esquisse dun programme. Readings mit opencourseware free online course materials. Moduli spaces of stable quotients and wallcrossing phenomena. Lecture notes mit opencourseware free online course materials. Pdf moduli spaces for torsion free modules on curve. The construction of moduli spaces and geometric invariant. Moduli spaces often carry natural geometric and topological structures as well. Pdf m theory moduli spaces and torsionfree structures.
In mathematics, there are many geometric and topological objects whose points are not actually points zero dimensional object rather they can represent different things. That is, the points of a moduli space correspond to solutions of geometric problems. Mirror symmetry for moduli spaces of higgs bundles via padic integration 3 there exist families of abelian varieties p 1. As an application, we prove in the case of a mukai vector of odd rank that they are birational to hilbert schemes. Its cohomology ring has a elegant structure, and mumford suggested studying the moduli space of curves in the same way. Concerning the geometry of moduli spaces of sheaves of higher rank, there are two guiding principles for the investigation. A general expression for the generating function of the euler characteristics of such moduli spaces is obtained.
We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of bridgeland. Clay mathematics proceedings volume 14 grassmannians. Oort, subvarieties of moduli spaces oor74 norman and oort, moduli of abelian varieties no80 b newton polygon strata. Nevertheless, something is known about the type of singularities they can attain. We introduce the moduli space of genus, pointed curves, with enough information.
Suppose x is a surface, e 0 is torsion free, degp 1. A major discovery of the 20th century, attributed to p. Kahler metric and moduli spaces, volume 18ii covers survey notes from the expository lectures given during the seminars in the academic year of 1987 for graduate students and mature mathematicians who were not experts on the topics considered during the sessions about partial differential equations. However, very little is known about this moduli space or its. In this paper, we study the cohomology of moduli spaces of gieseker semistable sheaves mx,h. But there is also the notion of coarse moduli space, which is a kind of conceptual hack designed to be able to keep thinking about what really wants to be a stack still as a plain sheaf.
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