Some notes about the construction of the hilbert scheme. The aim of the conference is to gather experts in various e. Two examples aregiven when this 2formis symplectic. Given a concrete category c, with objects denoted objc, and an equivalence relation on objc given by morphisms in c. Such moduli spaces have two di erent branches, named higgs branch and coulomb branch, joined at the origin, which in. Thus we have positive structures on these moduli spaces. That is, the points of a moduli space correspond to solutions of geometric problems. In this paper we give an overview of many instances of this phenomenon and in some cases relate it to predictions from theoretical physics.
Sometimes there may be spaces but this is more often what you might see in the pdf text. Minicourse on moduli spaces university of michigan. Moduli spaces of riemannian metrics wilderich tuschmann. For a clear introduction to stacks, i like martin olssons book compactifying moduli spaces for abelian varieties.
Generating functions for invariants of moduli spaces in algebraic geometry of are often related to modular forms. This provides a different approach to compactifying these spaces than. Weilpetersson volume of moduli spaces, mirzakhanis recursion and matrix models. Many pdf files do not actually contain any text spaces. An introduction to the topology of the moduli space of. X s be a flat, projective morphism of kschemes of finite type with integral geometric fibers. New moduli spaces of pointed curves and pencils of. Free points corresponding to part of s equal to 0 are also permitted. The examples include euler numbers of moduli spaces of sheaves on. 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. The objective of this thesis is to study the moduli spaces of pairs of mirror theories in 3 dimensions with n 4. Several topics in these notes will not be covered and some may be discussed only brie y in the.
As an application to global galois representations, we prove a theorem on the modularity comparing a deformation ring and a hecke ring. On the connected components of moduli spaces of finite. In gromovwitten theory, the domain curve is however required to vary in moduli thus acquiring nodes. Among other more elementary examples to set the stage, these two families of moduli spaces are surveyed in the next section. If n 2 then there is a unique complex structure i on msuch that gis a k ahler metric with respect to i. Schmitt, who studied the case when x is a nodal curve. The total space and the base space both being contractible, they concluded that diff1fand hence any other connected component of difff is contractible. Moduli spaces are spaces of solutions of geometric classification problems.
Here different solutions are identified if they are isomorphic that is, geometrically the same. The theorys principal application is to the construction of various moduli spaces. Cornalba, on the projectivity of the moduli spaces of curves, november 1992 page 2 that. To appreciate the distinction, show that p1 c is a ne moduli space for one dimensional. After a brief introduction to supersymmetry we will first focus on the higgs branch. Hyperpolygon spaces and moduli spaces of parabolic higgs bundles. In mathematics, in particular algebraic geometry, a moduli space is a geometric space whose. Orlicz spaces and modular spaces lecture notes in mathematics. On the projectivity of the moduli spaces of curves 1. 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. Curves of genus g, onedimensional subspaces of rn, finite sets. We show that mirzakhanis recursions for the volumes of moduli space of riemann surfaces are a special case of. The irreducibility of the space of curves of given genus pdf.
We prove that their transition functions are subtraction free. Explore free books, like the victory garden, and more browse now. This book studies certain spaces of riemannian metrics on both compact and. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. In this paper we only consider moduli spaces of objects on surfaces. 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. An introduction to moduli spaces of curves and its. Pdf on the geometry of moduli spaces of coherent systems. Given an ntuple of positive real numbers \alpha we consider the hyperpolygon space x\alpha, the hyperkahler quotient analogue to. Enter your mobile number or email address below and well send you a link to download the free kindle app. Then you can start reading kindle books on your smartphone, tablet, or computer no kindle device required. Msri dynamics on moduli spaces of geometric structures.
L is ample on z, and to do this it will be enough to check that the hypotheses of seshadris criterion of ampleness cf. Compactifying moduli spaces for abelian varieties martin c. We prove that the nonordinary component is connected in the moduli spaces of finite flat models of twodimensional local galois representations over finite fields. Moduli spaces from a topological viewpoint 387 map of the diff1f action, sctf tf locally has a section, so that. 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. 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. The metric on the moduli spaces of vortices on compact riemann surfaces where the. Let us call the closures of these strata the 1curves. Let x be a smooth algebraic surface over the complex field and let %. 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. Rational selfmaps of moduli spaces 337 applying this to the sheaf f l. Underthe same assumptions, the moduli spaces turn out to be irreducible. Observe that a coarse moduli space, if it exists, is unique up to isomorphism.
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. A good motivation for studying stacks is mumfords beautiful paper picard groups of moduli problems. Moduli spaces of hyperk ahler manifolds and mirror symmetry. Birational geometry of moduli spaces of sheaves and bridgeland stability jack huizenga abstract. Pixton 81 conjectured a beautiful formula for drg,s. 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. In dimension one, exactly one factor in the product of moduli spaces is m0. While the moduli space of stable maps remains virtually smooth, the quot compacti cation does not.
Suppose that we want to classify some kind of object, for example. It is proven that the composition of the yoneda coupling with the semiregularity map is a closed 2formon moduli spaces of sheaves. Minicourse on moduli spaces emily clader june 2011 1 what is a moduli space. This work builds on and is a natural continuation of our previous paper on the git compactification of the moduli space of cubic fourfolds. Invariants of moduli spaces and modular forms openstarts. Pdf weilpetersson volume of moduli spaces, mirzakhanis. Moduli spaces can be thought of as giving a universal space of parameters for the problem. The moduli spaces of holomorphic maps elenynicoleta ionel abstract this is a draft of some informal notes that are meant to accompany the second week of the graduate lectures at the 2012 ias women and mathematics program.
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. On the rationality of moduli spaces of vector bundles on fano. The necoarse moduli spaces are endowed with a scheme structure and carry more subtle information than the underlying set parameterizing the geometric objects. Let k be an algebraically closed field of characteristic zero. I will talk about a method of constructing virtual fundamental cycles on moduli spaces of jholomorphic maps.
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. Lecture notes mit opencourseware free online course. Moduli spaces of higgs bundles in mathematics and physics. Moduli spaces london mathematical society lecture note. We have 1 genuine space but lots of gaps between characters and some of those are actually spaces. 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. It was a masterful and understandable exposition of important material, with clear proofs and. 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. It follows then that the git compactification of the moduli space of cubic fourfolds is isomorphic to the looijengas compactification associated to this arrangement. For moduli i would second geometry of algebraic curves, volume ii. Stanford university, department of mathematics, building 380, stanford, california 94305, usa. They contain gaps between letters and the software has to guess if there is a space in the text. Moduli spaces of hyperk ahler manifolds and mirror symmetry 189 1 introduction let m.
484 731 387 1256 1127 665 1094 218 551 1034 1559 831 375 1468 1238 608 591 599 49 411 900 65 605 675 929 1025 1444 438 717 1289 1291 909 1299 1125 104 975 350 246 1032 1095 809 577 731