Calabiyau moduli schemes and moduli stacks pantev et al. Pdf improved converse theorems and fractional moduli of. Let mg,l be the moduli space of stable algebraic curves of genus g with l marked points. In dimension one, exactly one factor in the product of moduli spaces is m0. Beauville and donagishowedthat it is symplectic but did not construct an explicit symplectic form on it. Moduli spaces from a topological viewpoint 387 map of the diff1f action, sctf tf locally has a section, so that. Sloganhope start from the homotopy class of an isolated hypersurface singularity. Ag0402098 v3 1 mar 2004 moduli spaces and formal operads f. On the projectivity of the moduli spaces of curves 1. If is a moduli space of a certain kind of object or structure, then facts about can be interpreted as universal statements about all families of such objects. Bundles over moduli spaces and the quantisation of bps. The theorys principal application is to the construction of various moduli spaces. Since m higgs is a cotangent bundle, it carries a holomorphic symplectic form. These classes do not correspond to ordinary line bundles, but only to fractional ones.
The orlicz space l mt is re exive if and only if the nfunction m and its complementary function n both satisfy the 2condition 27, p. Find all the books, read about the author, and more. Introduction to moduli problems and orbit spaces subject. Intersection theory on the moduli space of curves ihes. We focus on sheaf theoretic compacti cations via quot schemetype constructions. There are further technical conditions that are omitted for the sake of brevity. Some quot schemes in tilted hearts and moduli spaces of stable pairs. But at the end of the day, they are only one of the techniques that can can be used to obtain information about the symplectic and contact structures.
Monopole operators, moduli spaces and dualities in 3d cs. The moduli spaces underlying the open tcft operad are those of stable riemann surfaces with boundary and marked points on the boundary. Our aim is to understand the topological nature of. For a clear introduction to stacks, i like martin olssons book compactifying moduli spaces for abelian varieties.
Moreover, we assume that p 1 and p 2 are dual in the sense of abelian varieties. A p 2, which act faithfully and transitively on the bres of m i. The moduli spaces of perturbed holomorphic maps have a beautiful and rich structure, not yet fully understood. 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. In this thesis, we discuss the charge 3 moduli space, calculate its metric and. In general, the moduli space m g for a surface f g of genus g is a complicated object. Marked singularities, their moduli spaces and atlases of stokes data claus hertling universit at mannheim 07. The key feature of the space rp1, considered as the moduli space of lines in the plane, is that it captures the ways in which lines can modulate, or vary continuously in families. I added a few words outlining other work of harer on the stability of the cohomology of the moduli space of curves. That is, the points of a moduli space correspond to solutions of geometric problems.
Both of them are moduli spaces of torsion sheaves on the cubic 4fold y. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. Inparticular, cohomology classes of the moduli space are universal cohomology classes. The moduli space of curves and its tautological ring by ravi vakil in the notices of the ams. Picard groups of moduli spaces, referring to 12, 6, and 2 for more details. There are at least three good reasons to study moduli spaces of sheaves on surfaces. In algebraic geometry, a moduli space of algebraic curves is a geometric space typically a scheme or an algebraic stack whose points represent isomorphism classes of algebraic curves. An interesting feature is that the moduli space of charge 3 lumps is a 7dimensional manifold of cohomogeneity one. An introduction to the moduli spaces of curves by maarten hoeve.
Schmitt, moduli spaces for principal bundles in moduli spaces and vector bundles, london math. Marked singularities, their moduli spaces and atlases of. Monopole operators, moduli spaces and dualities in 3d cs matter theories mauricio romo university of california, santa barbara 2011 d. Formal definitions and generalities related to various marked moduli spaces are to be found in section 1. For a nice discussion of the rationality of moduli spaces of curves of low genus, see dolgachevs article do. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. The total space and the base space both being contractible, they concluded that diff1fand hence any other connected component of difff is contractible. In particular this yields new proofs of ribbon graph complexes computing the homology of these moduli spaces. Here we emphasize the crucial role played by l x1 h, h h closed surface, in connection with. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. For two bounded surfaces m the oneholed torus research supported under nsf grant dms0103889. Orlicz spaces and modular spaces lecture notes in mathematics 1983rd edition by julian musielak author visit amazons julian musielak page. Observe that a coarse moduli space, if it exists, is unique up to isomorphism.
The base space of a certain global versal unfolding should be an. Let us call the closures of these strata the 1curves. Numerous and frequentlyupdated resource results are available from this search. A good motivation for studying stacks is mumfords beautiful paper picard groups of moduli problems.
Improved converse theorems and fractional moduli of smoothness in orlicz spaces 53 and equivalently, for r 0 and t 0. Curve complexes and the dmcompactification of moduli spaces. The aim of the conference is to gather experts in various e. Here different solutions are identified if they are isomorphic that is, geometrically the same. Since the moduli space is a point, these must induce the same map to the moduli space, despite being nonisomorphic. For moduli i would second geometry of algebraic curves, volume ii.
Tikhonov, on moduli of smoothness of fracional order, real ana. Tikhonov, moduli of smoothness and the interrelation of some classes of functions, function spaces, interpolation theory and related topics lond. We discuss the moment of inertia or angular integral of moduli spaces of charge 3 lumps. Preliminary reduction steps let x be an artin stack, and x ian open covering of x by open substacks for which there exist coarse moduli spaces. For example, the classifying space of rank complex vector. Moduli spaces can be thought of as giving a universal space of parameters for the problem. To such data, we associate an explicit meromorphic higgs bundle on a genusg riemann surface, where g is the number of loops in the comet. Some notes containing a brief survey of the cohomology of the moduli space of curves and the harerzagier formula for the orbifold euler characteristic.
Pdf moduli structures, separability of the kinematic. The irreducibility of the space of curves of given genus numdam. Mirror symmetry for moduli spaces of higgs bundles via padic integration 3 there exist families of abelian varieties p 1. The necoarse moduli spaces are endowed with a scheme structure and carry more subtle information than the underlying set parameterizing the geometric objects. 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 related to. Moduli spaces of higgs bundles in mathematics and physics. Several important moduli spaces are described by cubic surfaces in a. To appreciate the distinction, show that p1 c is a ne moduli space for one dimensional. Rational points on moduli spaces of curves by dave jensen. The objective of this thesis is to study the moduli spaces of pairs of mirror theories in 3 dimensions with n 4. Moduli spaces of quasimaps to projective space with perfect obstruction theories younghoon kiem abstract. Indeed, a family of onedimensional vector spaces is a line bundle, and over the base b s1, there are two nonisomorphic line bundles. Some notes describing the picard group of the moduli space of curves. Moduli spaces of stable quotients and wallcrossing phenomena.
We wish to consider the corresponding theory for ktfts. The symmetries of the original moduli problem determines a group of transformations of the moduli space which generates a dynamical system. It turns out that the generic ber of the hitchin map h is a. Yan soibelman kansas state university moduli spaces of higgs bundles in mathematics and physicsnovember 19, 20 14 33. Page 1 geometricquantizationfor themodulispaceofvector. 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. Recently the structure of these spaces has been put to good use in understanding several moduli spaces of solutions to variational problems, such as the moduli of holomorphic maps of surfaces into certain complex manifolds, the moduli of instantons, and the chow varieties. After a brief introduction to supersymmetry we will first focus on the higgs branch. Stacks let sbe a scheme, and let s schs be the category of schemes over s. Li that quasimaps of degree d0 to pn 1 over mpointed curves of genus g form an algebraic. Two examples aregiven when this 2formis symplectic. 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. Simpson, moduli of representations of the fundamental group of a smooth projective variety, i, inst. A category over sis a category f together with a covariant functor p f.
Sylvain delpech asymptotic uniform moduli and kottman constant 3. Moduli spaces are spaces of solutions of geometric classification problems. It was a masterful and understandable exposition of important material, with clear proofs and. Asymptotic uniform moduli and kottman constant of orlicz. Ingeometryoneusuallywantstodefineinvariantsof thespaces,orgeometricobjects,underconsideration. Underthe same assumptions, the moduli spaces turn out to be irreducible. The setting of orlicz sequence spaces we refer to 8 for all the background about orlicz sequence spaces. Hitchin integrable system i will assume for simplicity that m higgs is a manifold. Moreover, the study of moduli spaces has benefited tremendously in recent years from interactions with physics in.
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