Pdf m theory moduli spaces and torsionfree structures. Lecture notes mit opencourseware free online 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. Find materials for this course in the pages linked along the left. A moduli space for finitely generated torsion free rmodules of rank 1 is constructed by g. Moduli spaces can be thought of as giving a universal space of parameters for the problem. Torelli groups and geometry of moduli spaces of curves. Moduli spaces london mathematical society lecture note. The argument makes use of a new chowtheoretic result, showing that moduli spaces on an enriques surface.
Moduli spaces from a topological viewpoint ib madsen abstract. An introduction to the topology of the moduli space of. 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. In this paper we relate two important but disparate topics in the. The lecture notes section lists the lecture topics, along with the lecture notes files, covered in the course. Moduli definition of moduli by the free dictionary. Curves of genus g, onedimensional subspaces of rn, finite sets. Euler characteristics of moduli spaces of torsion free. 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. Suppose x is a surface, e 0 is torsion free, degp 1.
Underthe same assumptions, the moduli spaces turn out to be irreducible. A p 2, which act faithfully and transitively on the bres of m i. Topological moduli spaces of knots cornell university. Birational geometry and moduli spaces elisabetta colombo. Good geometry of the moduli spaces of riemann surfaces. Its importance was emphasized by grothendieck in his famous esquisse dun programme. In mathematics, there are many geometric and topological objects whose points are not actually points zero dimensional object rather they can represent different things. Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces. Using wallcrossing for k3 surfaces, we establish birational equivalence of moduli spaces of stable objects on generic enriques surfaces for different stability conditions. You can read online an introduction to riemann surfaces algebraic curves and moduli spaces here in pdf, epub, mobi or docx formats. 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. Here different solutions are identified if they are isomorphic that is, geometrically the same.
One can consider moduli spaces of vector bundles, coherent sheaves, etc. 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. 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. 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. So the term fine moduli space mainly indicates that a given object that might be a delignemumford stack is actually just a plain scheme. Suppose that we want to classify some kind of object, for example. Devadoss introduction the riemann moduli space mn gof surfaces of genus gwith nmarked points has become a central object in mathematical physics. Yoshioka computes the betti numbers of moduli spaces of rank 2 sheaves on p2 and proves the. The construction of moduli spaces and geometric invariant theory by dinamo djounvouna in algebraic geometry, classi.
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. Compact moduli spaces of surfaces and exceptional vector bundles. M theory moduli spaces and torsionfree structures mariana grana, c. 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. A general expression for the generating function of the euler characteristics of such moduli spaces is obtained. 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 boardman vogt resolution and tropical moduli spaces. The aim of this paper is to show how facts about the homology of the torelli group imply. 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. That is, the points of a moduli space correspond to solutions of geometric problems.
Moduli spaces are spaces of solutions of geometric classification problems. For instance, if g is a linearly reductive group scheme acting linearly on x. An example of a different kind is the jacobian, which is a moduli space that classi. Nevertheless, something is known about the type of singularities they can attain. Torelli groups and geometry of moduli spaces of curves richard m. The moduli space is, therefore, the positive real numbers. Clay mathematics proceedings volume 14 grassmannians. 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. The most basic problem is that of moduli of smooth complete curves of a fixed genus.
This textbook offers a concise yet thorough introduction to the notion of moduli spaces of complex algebraic curves. Pdf mtheory moduli spaces and torsionfree structures. The generating function is expressed in terms of euler characteristics of certain moduli spaces of stable configurations of linear subspaces appearing in classical git. The most prominent one is the moduli space mg of all riemann surfaces of genus g. We shall construct a moduli space of pairs of k\ahlereinstein structures and special lagrangians and obtain smoothness of the moduli space of these pairs.
An introduction to moduli spaces of curves and its. 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. Localizations on moduli spaces and free field realizations of feynman rules jian zhou abstract. Msri series volume 28, 1995 torelli groups and geometry of moduli spaces of curves richard m. The purpose of this article is to define and study moduli spaces of colored graphs in the spirit of and where such spaces for uncolored graphs were used to study the homology of automorphism groups of free groups. Minicourse on moduli spaces university of michigan.
An introduction to the topology of the moduli space of stable. Architecture form, space, & order third edition architecture form, space, & order third edition. Geometry of moduli spaces and representation theory. We also compare these real moduli spaces to the real points of the corresponding complex. 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. 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. Some notes containing a brief survey of the cohomology of the moduli space of curves and the harerzagier formula for the orbifold euler characteristic. We also describe singularities and some other features of it. Abstract references similar articles additional information. In this paper, we study the cohomology of moduli spaces of gieseker semistable sheaves mx,h. The study of moduli spaces is an old branch of algebraic geometry with. We introduce the moduli space of genus, pointed curves, with enough information. A copy of the license is included in the section entitled gnu free documentation license.
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. Moduli spaces of stable quotients and wallcrossing phenomena. A major discovery of the 20th century, attributed to p. Dynamics of teichmuller modular groups and topology of moduli. Topology of moduli spaces of free group representations in real reductive groups. 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. 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. Its cohomology ring has a elegant structure, and mumford suggested studying the moduli space of curves in the same way. Permission is granted to copy, distribute andor modify this document under the terms of the gnu free documentation license, version 1. Moduli spaces often carry natural geometric and topological structures as well. Birational geometry of moduli spaces of stable objects on.
Moduli spaces of stable vector bundles on enriques surfaces. A p 2, which act faithfully and transitively on the bres of. Minicourse on moduli spaces emily clader june 2011 1 what is a moduli space. The book begins by studying individual smooth algebraic curves, including the most beautiful ones, before addressing families of curves. Using the simplest methods, and proving as much as possible, formulas are derived for the betti numbers. 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. 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. One also distinguishes between fine and coarse moduli spaces for the same moduli problem.
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. As an application, we prove in the case of a mukai vector of odd rank that they are birational to hilbert schemes. We prove iqbals conjecture on the relationship between the free energy of closed string theory in local toric geometry and the wesszuminowitten model. 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.
We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of bridgeland. Moreover, we assume that p 1 and p 2 are dual in the sense of abelian varieties. Further we show that the moduli space of these pairs is locally embedded in a certain relative cohomology group. Good moduli spaces appear to be the correct notion characterizing morphisms from stacks arising from quotients by linearly reductive groups to the quotient scheme. 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. Mirror symmetry for moduli spaces of higgs bundles via padic integration 3 there exist families of abelian varieties p 1. Introduction let x be an artin stack always assumed to have quasicompact and separated diagonal over specz. 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. There are further technical conditions that are omitted for the sake of brevity. Readings mit opencourseware free online course materials. However, very little is known about this moduli space or its.
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. Moduli spaces of special lagrangians and kahlereinstein. Download book an introduction to riemann surfaces algebraic curves and moduli spaces in pdf format. Moduli spaces of sheaves of rank 2 on a surface are not smooth, unless we consider sheaves with special invariants on special surfaces. Lecture notes mit opencourseware free online course materials. This divisor class varies naturally with the bridgeland stability condition. Mcmullen proved that the moduli spaces of riemann surfaces are k. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. 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. Concerning the geometry of moduli spaces of sheaves of higher rank, there are two guiding principles for the investigation. Pdf moduli spaces for torsion free modules on curve. 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. The construction of moduli spaces and geometric invariant.
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