Lecture notes mit opencourseware free online course. In mathematics, in particular algebraic geometry, a moduli space is a geometric space whose. Besides the boundary divisors, we have the divisor classes i c1li, where li is the line bundle with ber the cotangent space to cat xi, and, the rst chern class of the hodge bundle e of rank g. Let x be a smooth algebraic surface over the complex field and let %. Rational selfmaps of moduli spaces 337 applying this to the sheaf f l. Free points corresponding to part of s equal to 0 are also permitted. An introduction to the topology of the moduli space of. Such moduli spaces have two di erent branches, named higgs branch and coulomb branch, joined at the origin, which in. Newstead gave a series of lectures in 1975 at the tata institute of fundamental research, mumbai, on git and its application to the moduli of vector bundles on curves. That is, the points of a moduli space correspond to solutions of geometric problems. Suppose that we want to classify some kind of object, for example. Among other more elementary examples to set the stage, these two families of moduli spaces are surveyed in the next section.
The aim of the conference is to gather experts in various e. Moduli spaces from a topological viewpoint 387 map of the diff1f action, sctf tf locally has a section, so that. In this paper we give an overview of many instances of this phenomenon and in some cases relate it to predictions from theoretical physics. Curves of genus g, onedimensional subspaces of rn, finite sets. Explore free books, like the victory garden, and more browse now. Enter your mobile number or email address below and well send you a link to download the free kindle app. The construction uses quite a bit of homological algebra, in particular homotopy colimits and homotopy sheaves, and most of the action happens at the chain level. Given an ntuple of positive real numbers \alpha we consider the hyperpolygon space x\alpha, the hyperkahler quotient analogue to. As an application to global galois representations, we prove a theorem on the modularity comparing a deformation ring and a hecke ring. The irreducibility of the space of curves of given genus pdf. Moduli spaces can be thought of as giving a universal space of parameters for the problem. To appreciate the distinction, show that p1 c is a ne moduli space for one dimensional.
The original conjecture of 3d mirror symmetry was motivated by the fact that in these pairs of theories the higgs and coulomb branches are swapped. This book studies certain spaces of riemannian metrics on both compact and. Let k be an algebraically closed field of characteristic zero. The objective of this thesis is to study the moduli spaces of pairs of mirror theories in 3 dimensions with n 4. Msri dynamics on moduli spaces of geometric structures. For a clear introduction to stacks, i like martin olssons book compactifying moduli spaces for abelian varieties. After a brief introduction to supersymmetry we will first focus on the higgs branch. It follows then that the git compactification of the moduli space of cubic fourfolds is isomorphic to the looijengas compactification associated to this arrangement.
Sometimes there may be spaces but this is more often what you might see in the pdf text. An introduction to moduli spaces of curves and its. Moduli spaces london mathematical society lecture note. Then you can start reading kindle books on your smartphone, tablet, or computer no kindle device required. X s be a flat, projective morphism of kschemes of finite type with integral geometric fibers. Some notes about the construction of the hilbert scheme. A good motivation for studying stacks is mumfords beautiful paper picard groups of moduli problems. The examples include euler numbers of moduli spaces of sheaves on. Pixton 81 conjectured a beautiful formula for drg,s. Hyperpolygon spaces and moduli spaces of parabolic higgs bundles. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions.
Given a concrete category c, with objects denoted objc, and an equivalence relation on objc given by morphisms in c. In dimension one, exactly one factor in the product of moduli spaces is m0. Minicourse on moduli spaces emily clader june 2011 1 what is a moduli space. Several topics in these notes will not be covered and some may be discussed only brie y in the. Two important examples are the moduli spaces of riemann surfaces and the closely related moduli spaces of at connections on principal gbundles over surfaces for some lie group g. Orlicz spaces and modular spaces lecture notes in mathematics. Many pdf files do not actually contain any text spaces. This provides a different approach to compactifying these spaces than. I will talk about a method of constructing virtual fundamental cycles on moduli spaces of jholomorphic maps. We prove the existence of a projective relative moduli space for semistable singular principal bundles on the fibers of f this generalizes the result of a.
The total space and the base space both being contractible, they concluded that diff1fand hence any other connected component of difff is contractible. We prove that the nonordinary component is connected in the moduli spaces of finite flat models of twodimensional local galois representations over finite fields. The purpose of this conference is to bring together mathematicians working in the field of moduli spaces of geometric and algebraic structures which might be. Minicourse on moduli spaces university of michigan. Compactifying moduli spaces for abelian varieties martin c. Given an f qlinear category such that the moduli space of its objects is a smooth artin stack and some additional conditions we give formulas for an exponential sum over the set of abso. Moduli spaces of sheaves and hilbert schemes of points have experienced a recent resurgence in interest in the past several years, due largely to new techniques arising from bridgeland stability conditions and derived category methods. We show that mirzakhanis recursions for the volumes of moduli space of riemann surfaces are a special case of. Birational geometry of moduli spaces of sheaves and bridgeland stability jack huizenga abstract. Thus we have positive structures on these moduli spaces. For moduli i would second geometry of algebraic curves, volume ii.
Cornalba, on the projectivity of the moduli spaces of curves, november 1992 page 2 that. The metric on the moduli spaces of vortices on compact riemann surfaces where the. Moduli spaces of higgs bundles in mathematics and physics. On the projectivity of the moduli spaces of curves 1. Underthe same assumptions, the moduli spaces turn out to be irreducible.
Let us call the closures of these strata the 1curves. Invariants of moduli spaces and modular forms openstarts. Yan soibelman kansas state universitymoduli spaces of higgs bundles in mathematics and physicsnovember 19, 20 18 33 hitchin integrable systems and donaldsonthomas theory i start with mathematics and later discuss the related physics. The theorys principal application is to the construction of various moduli spaces. If n 2 then there is a unique complex structure i on msuch that gis a k ahler metric with respect to i. Weilpetersson volume of moduli spaces, mirzakhanis recursion and matrix models. Pdf on the geometry of moduli spaces of coherent systems. On the connected components of moduli spaces of finite. L is ample on z, and to do this it will be enough to check that the hypotheses of seshadris criterion of ampleness cf. Moduli spaces are spaces of solutions of geometric classification problems.
Here different solutions are identified if they are isomorphic that is, geometrically the same. This work builds on and is a natural continuation of our previous paper on the git compactification of the moduli space of cubic fourfolds. It was a masterful and understandable exposition of important material, with clear proofs and. On the rationality of moduli spaces of vector bundles on fano. In this paper we only consider moduli spaces of objects on surfaces. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. New moduli spaces of pointed curves and pencils of. While the moduli space of stable maps remains virtually smooth, the quot compacti cation does not. Moduli spaces of hyperk ahler manifolds and mirror symmetry. Moduli spaces of riemannian metrics wilderich tuschmann. Stanford university, department of mathematics, building 380, stanford, california 94305, usa. Pdf weilpetersson volume of moduli spaces, mirzakhanis. Moduli spaces of hyperk ahler manifolds and mirror symmetry 189 1 introduction let m.
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