Moduli theory

A theory studying continuous families of objects in algebraic geometry.

Let be a class of objects in algebraic geometry (varieties, schemes, vector bundles, etc.) on which an equivalence relation has been given. The fundamental classification problem (the description of the set of classes ) has the following two parts: 1) the description of discrete invariants, which usually allow a partition of into a countable number of subsets, the objects of which already continuously depend on parameters; 2) the assignment and study of algebro-geometric structures on the parameter sets. The second part forms the matter of moduli theory.

Moduli theory arose in the study of elliptic functions: There is a continuous family of different fields of elliptic functions (or of their models — isomorphic elliptic curves over ), parametrized by the complex numbers. B. Riemann, who introduced the term "moduli" , showed that an algebraic function field over (or their models, compact Riemann surfaces) of genus depends on continuous complex parameters — the moduli.

Basic concepts in moduli theory.

Let be a scheme (a complex or algebraic space). A family of objects parametrized by the scheme (or, as is often said, "scheme over Sover S" or "scheme with basis Swith basis S" ) is a set of objects

equipped with an additional structure compatible with the structure of the base . This structure, in each concrete case, is given explicitly. A functor of families is a contravariant functor from the category of the schemes (or spaces) into the category of sets defined as follows: is the set of classes of isomorphic families over . To every morphism is associated a mapping which assigns to a family over the pullback, or induced, family over .

Let be an object in the category of schemes (complex or algebraic spaces) and let be a functor of the points in this category, that is, . If the functor of families is representable, that is, for some , then there exists a universal family with base , and is called a fine moduli scheme (respectively, fine complex moduli space or fine algebraic moduli space). The functor is representable in very few cases. Therefore the notion of a coarse moduli scheme was introduced. is called a coarse moduli scheme if there is a morphism of functors with the properties: a) if is one point (where is an algebraically closed field), then the mapping is bijective; in other words, the set of geometric points of the scheme is in a natural one-to-one correspondence with the set of equivalence classes of parametrized objects; and b) for each scheme and morphism of functors there is a unique morphism such that . Coarse schemes of complex and algebraic moduli spaces are similarly defined.

Although a coarse moduli scheme uniquely parametrizes the class of objects defined by given discrete invariants, the natural family over it (in contrast to the family over a fine moduli scheme) does not have the strong universality property. A coarse moduli scheme (space) already exists in a fairly large number of cases.

Examples. 1) Moduli of algebraic curves. Let (respectively ) be the set of classes of isomorphic projective non-singular curves (respectively, stable curves) of genus over an algebraically closed field . A family over is a smooth (flat) proper morphism of schemes whose fibres are smooth (stable) curves of genus . Then there is a coarse (but not a fine) moduli scheme (respectively ), which is a quasi-projective (projective), irreducible, normal variety over (see [3], [5], [6]).

2) Moduli of algebraic curves with level structure (with Jacobian rigidity). Let be a smooth family of projective curves (respectively, a flat family of stable curves) of genus , let be an integer invertible on , and let be the first direct image of the constant sheaf in the étale topology. Then is locally free, has rank and is equipped with a locally non-degenerate symplectic form with values in , up to an invertible element in . A Jacobian structure of level on is an assignment of a symplectic isomorphism

Let (respectively, ) be the functor of families of smooth (stable) curves of genus with a Jacobian rigidity of level . Then for the functor (respectively, ) is represented by a quasi-projective (projective) scheme (respectively, ) over , where is the inverse image of an -th root of unity, that is, there is a fine moduli scheme (respectively, ) for the smooth (stable) curves of genus over a field of characteristic coprime with , equipped with a Jacobian rigidity of level . For sufficiently large the scheme is smooth [5].

3) Polarized algebraic varieties. A polarized family is a pair , where is a smooth family of varieties, i.e. a smooth proper morphism , and is the class of the relatively ample invertible sheaf in modulo , where is the relative Picard scheme and is the connected component of its zero section. In this case a functor of the polarized families , with a given Hilbert polynomial , is constructed. Without additional restrictions this functor is not representable. The existence of a coarse moduli space is known (1989) only in individual cases.

For polarized algebraic varieties the idea of rigidity of level also exists.

4) Vector bundles. There are also results on moduli spaces for vector bundles of rank over an algebraic variety . In this case a family over is a vector bundle over . Cf. [7], [10]–[14] for a description of results and more detail.

Local and global theory.

The local theory arose as the theory of deformations of complex structures (see Deformation 1) and 2)). The fundamental methods of the global theory are those of the theory of representable functors and geometric invariant theory, the theory of algebraic stacks, and the algebraization of formal moduli.

The method of construction of a global moduli space goes back to the classical theory of invariants (cf. Invariants, theory of). It is as follows. A sufficiently large family is constructed which contains representatives of all equivalence classes of the objects in questions, and so that the equivalence relation on reduces to the action of an algebraic group . Then the theory of actions of algebraic groups on algebraic varieties (schemes, spaces) is exploited with the aim of clarifying conditions for the existence of the quotient in the corresponding category. The basic tool in the construction of the family is the theory of Hilbert schemes (cf. Hilbert scheme). In such an approach the difficulty in constructing the family is reduced to the problem on a simultaneous immersion of the objects in question into a projective space. An important result on the possibility of such a simultaneous immersion is Matsusaka's theorem. Then the difficult problem remains of the existence of the quotient . Here one has the notions of categorical and geometric quotients. The construction of a coarse moduli space reduces to the problem of the existence of geometric quotients; here the idea of stability of points, corresponding to the idea of orbits in general position, is used. Results concerning actions of reductive groups on algebraic varieties over fields of characteristic have been extended to the case of fields of characteristic .

Another approach to global moduli theory is the method of algebraic stacks, that is, a method of globalization of local deformation theory. The first step in the investigation of the representability of a global functor of families in this approach is the establishment of the algebraizability of a formal versal deformation for each object . The difficulty in the construction of a global moduli space is that not every factorization of the base of the family with respect to an equivalence relation is a separated space. In such cases the object representing the functor is replaced by an algebraic stack, the study of the properties of which gives some information on the moduli space.

One of the approaches to global moduli theory over is the theory of period mappings (cf. Period mapping). The fundamental object here is the classifying space of polarized Hodge structures (cf. Hodge structure) of weight for given Hodge numbers. For a family of polarized algebraic varieties over the periods define a mapping of onto the corresponding classifying space of Hodge structures. The moduli problem reduces to the study of conditions for the period mapping to be bijective. The presence of (global) injectivity for the period mapping is the so-called local-global Torelli problem. Along this route the existence of coarse moduli spaces has been proved for curves, Abelian varieties and -surfaces.

The compactification problem for a moduli variety is that of finding a natural and complete (projective or compact, in the theory over the field ) variety containing as a dense open subset, and also the description and geometric interpretation of the boundary . In example 1) the natural compactification of the coarse moduli variety of curves of genus is the projective moduli variety of stable curves. For polarized Abelian varieties over several means for compactifying moduli varieties are known.

References

[1]	M. Artin, "Algebraization of formal moduli 1" , Global analysis , Univ. Tokyo Press (1969) pp. 21–71
[2]	M. Artin, "Versal deformations and algebraic stacks" Invent. Math. , 27 (1974) pp. 165–189
[3]	P. Deligne, D. Mumford, "The irreducibility of the space of curves of given genus" Publ. Math. IHES , 36 (1969) pp. 75–109
[4]	J. Dieudonné, J.B. Carrell, "Invariant theory, old and new" , Acad. Press (1971)
[5]	D. Mumford, "Geometric invariant theory" , Springer (1965)
[6]	D. Mumford, "Stability of projective varieties" l'Enseign. Math. (2) , 23 : 1–2 (1977) pp. 39–110
[7]	D. Gieseker, "Global moduli for varieties of general type" Invent. Math. , 43 (1977) pp. 233–282
[8]	D. Gieseker, "On the moduli of vector bundles on an algebraic surface" Ann. of Math. , 106 (1977) pp. 45–60
[9]	T. Matsusaka, "Polarized varieties with a given Hilbert polynomial" Amer. J. Math. , 94 : 4 (1972) pp. 1027–1077
[10]	P.E. Newstead, "Lectures on introduction to moduli problems and orbit spaces" , Springer (1978)
[11]	C. Okonek, M. Schneider, H. Spindler, "Vector bundles on complex projective spaces" , Birkhäuser (1980)
[12]	H. Popp, "Moduli theory and classification theory of algebraic varieties" , Springer (1977)
[13]	C.S. Seshadri, "Spaces of unitary vector bundles on a compact Riemann surface" Ann. of Math. , 85 (1967) pp. 302–336
[14]	A.N. Tyurin, "The geometry of moduli of vector bundles" Russian Math. Surveys , 29 : 6 (1974) pp. 57–88 Uspekhi Mat. Nauk , 29 : 6 (1974) pp. 59–88

Comments

Much progress has been made recently in the study of the moduli spaces of algebraic curves and Abelian varieties; see the appendices of [a2]. Among the most important ones are the question of the compactification of (see the appendix to Chapt. 5 in [a1], which is an enlarged edition of [5]) and the proof that is of general type for [a2].

G. Faltings has constructed a compactification of the moduli space of principally-polarized Abelian varieties over , see [a3].

References

[a1]	D. Mumford, J. Fogarty, "Geometric invariant theory" , Springer (1982)
[a2]	J. Harris, "Curves and their moduli" S.J. Bloch (ed.) , Algebraic geometry , Proc. Symp. Pure Math. , 46.1 , Amer. Math. Soc. (1985) pp. 99–143
[a3]	G. Faltings, "Arithmetische Kompaktifizierung des Modulraums der abelschen Varietäten" F. Hirzebruch (ed.) J. Schwermer (ed.) S. Suter (ed.) , Arbeitstagung Bonn 1984 , Lect. notes in math. , 1111 , Springer (1985) pp. 321–383