##
**Quasi-projective moduli for polarized manifolds.**
*(English)*
Zbl 0844.14004

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 30. Berlin: Springer-Verlag. viii, 320 p. (1995).

The word “moduli” is due to B. Riemann, who proved in his celebrated paper of 1857 on abelian functions that an isomorphism class of Riemann surfaces of genus \(g \geq 2\) depends on \(3g - 3\) continuously varying complex parameters, which he called the “moduli” of this class. More generally, a “moduli problem” is associated with any specific class of algebro-geometric objects (e.g., varieties, sheaves on a variety, etc.), together with a certain equivalence relation on it, and with a suitably defined notion of “families of objects varying algebraically with respect to a parameter scheme”. The “moduli space” – if it exists – for a given moduli problem is an algebraic scheme, whose points are in one-to-one correspondence with the equivalence classes of objects occurring in the moduli problem, and which is, in a certain sense, a “universal” parameter scheme with respect to all families of objects. Moduli theory, as a part of classification theory in algebraic geometry, therefore deals with two kinds of basic questions:

(1) Is there a moduli space for a given moduli problem (existence and construction of moduli spaces)?

(2) If so, what are the geometric properties of that moduli space (e.g., dimension, singularity or non-singularity, connectedness, projectivity, quasi-projectivity, (co-)homological invariants, Chow groups, (uni-)rationality, ample divisors, etc.)?

Although moduli problems emerged rather early in the history of algebraic geometry, a rigorous formulation of the right concept for them, just in the abstract sense mentioned above, as well as the precise definition of what a moduli scheme should be, has only been given pretty recently. Elaborating the pioneering ideas of A. Grothendieck, and combining them with the framework of invariant theory, in a just as ingenious a manner, D. Mumford established modern moduli theory and geometric invariant theory in the early 1960s. Since the appearance of D. Mumford’s fundamental monograph “Geometric invariant theory” (1965; Zbl 0147.39304), in which he provided the conceptual and methodical principles of algebro-geometric moduli theory, together with an explicit construction of (coarse) moduli schemes for some concrete classes of objects (e.g., smooth algebraic curves, polarized abelian varieties), there has been an explosion of activity in investigating various moduli problems in algebraic geometry. The third, enlarged edition of Mumford’s famous “GIT” [cf.: D. Mumford, J. Fogarty and F. Kirwan, “Geometric invariant theory”, 3rd edition (1993; Zbl 0797.14004)] gives, in seven updated appendices, a good survey on the developments over the past thirty years. – In attempting to show the existence of moduli schemes for concrete moduli problems (via geometric invariant theory), one often has to verify that the quotient of a scheme by a group action is again a scheme, or at least an algebraic space (in the sense of M. Artin). Once having been able to construct a moduli scheme in this way, the question of whether it is (quasi-)projective leads to the problem of constructing ample sheaves on it.

This general approach to constructing moduli schemes, together with a projective embedding, is the central theme of the present book under review. Roughly speaking, the author discusses two subjects of seemingly quite different nature. The first one concerns construction methods for good quotients of quasi-projective schemes modulo group actions (or modulo equivalence relations, respectively), while the second subject of the text incorporates the study of the direct images of higher powers of dualizing sheaves associated with families of complex projective varieties.

The methods developed in the course of the discussion of these two topics are then combined in order to construct quasi-projective moduli spaces for certain types of moduli problems. This approach to a general construction principle for moduli spaces is motivated by (and based upon) the author’s earlier, extensive work on weakly positive sheaves associated with algebra fibre spaces, and their application to the study of Iitaka’s “\(C^+_{n,m}\)-conjecture” [cf.: E. Viehweg, in: Algebraic Varieties and Analytic Varieties, Proc. Symp, Tokyo 1981, Adv. Stud. Pure Math. 1, 329-353 (1983; Zbl 0513.14019)]. The link between the “\(C^+_{n,m}\)-conjecture”, the property of weak positivity for direct image sheaves, and moduli theory for complex manifolds of general type has led the author to investigate this interrelation more closely, and to use “weak positivity” and geometric invariant theory, in a general way, to construct suitable moduli problems and, simultaneously, quasi-projective coarse moduli spaces for them. This has been published in a series of papers which appeared between 1989 and 1991 [cf. H. Esnault and E. Viehweg, in: Algebraic Geometry and analytic Geometry, Proc. Conf., Tokyo 1990, ICM-90 Satell. Conf. Proc., 53-80 (1991; Zbl 0772.14009); E. Viehweg, “Weak positivity and the stability of certain Hilbert points”, I, II, and III, Invent. Math. 96, No. 3, 639-667 (1989; Zbl 0695.14006) and 101, No. 1, 191-223 (1990; Zbl 0721.14007) and No. 3, 521-543 (1990; Zbl 0746.14014); and E. Viehweg, Math. Ann. 289, No. 2, 297-314 (1991; Zbl 0729.14010)].

The present monograph is a systematic, comprehensive and textbook-like elaboration of the author’s work published in these papers. Compared to them, the author has generalized some results (e.g., the stability criteria in chapter 4), simplified many proofs and constructions, and he has given the entire presentation a nearly self-contained character by including (with proofs) most of those results which are perhaps well-known to specialists but barely documented in the standard texts on algebraic geometry.

Chapter 1 provides therefore the basic material from moduli theory: moduli problems, moduli functors, coarse and fine moduli schemes, Grothendieck’s construction of Hilbert schemes, and Hilbert schemes of (canonically) polarized schemes. – Chapter 2 deals with numerically effective and weakly positive sheaves, including the author’s covering constructions and generalizations of the Fujita-Kawamata positivity theorem. The basic definitions and results from Mumford’s geometric invariant theory are compiled in chapter 3, while chapter 4 turns to the author’s approach to constructing stable points in Hilbert schemes by means of weakly positive sheaves and the ampleness criteria of Viehweg and Kollár. Chapter 5 interrupts the program of constructing moduli schemes of canonically polarized manifolds a little bit. Auxiliary results on locally free sheaves and divisors, including an unpublished theorem of O. Gabber on extensions of certain locally free sheaves to compactifications, a discussion of singularities of divisors (in flat families), and some vanishing theorems, form the heart of this more technical chapter. It is, however, very important in the sequel, when moduli schemes also for non-canonically polarized manifolds are to be constructed. – Chapter 6 presents the author’s methods and results on weakly positive direct image sheaves, as they can be found in his original papers. Using O. Gabber’s extension theorem, the author presents the material in a very elegant, simplified manner. This is then used, in chapter 7, to construct moduli schemes and, more generally, geometric quotients. The ampleness criteria from chapter 4 even provide quasi-projective geometric quotients. In order to establish the existence of moduli schemes for certain classes of singular varieties, the author discusses (at the end of chapter 7) the conditions for that, in the case of larger classes of moduli functors. – Chapter 8 turns then to moduli functors of normal varieties with canonical singularities. It is shown that, under additional assumptions (boundedness, local closedness, separatedness) for such functors, the method of constructing quasi-projective moduli schemes (developed so far) carries over this case, too, including even non-canonically polarized varieties. Chapter 9, the concluding chapter, discusses another approach of constructing moduli schemes, namely the one via M. Artin’s theory of algebraic spaces (or, in the complex case, via the theory of Moishezon spaces). After a brief introduction to this framework, the author explains how to construct quotients by equivalence relations in the category of algebraic spaces, and to prove then algebraicity and quasi-projectivity by methods developed by himself and by J. Kollár [J. Differ. Geom. 32, No. 1, 235-268 (1990; Zbl 0684.14002)]. – The author’s general construction principle for quasi-projective moduli schemes re-establishes most of the known existence theorems for curves, abelian varieties, surfaces of general type, \(K\)-3 surfaces, etc., and that in a unified way. However, in some cases, the ample line bundles that he uses to prove quasi-projectivity are less nice and explicit as in the constructions given before.

Altogether, the book under review is an outstanding research monograph on moduli theory. It presents the subject and its very recent achievements in a most general, systematic and comprehensive manner. As detailed and self-contained as it is, the text may be regarded as a worthy, contemporary successor of D. Mumford’s celebrated “Geometric invariant theory” (loc. cit.).

(1) Is there a moduli space for a given moduli problem (existence and construction of moduli spaces)?

(2) If so, what are the geometric properties of that moduli space (e.g., dimension, singularity or non-singularity, connectedness, projectivity, quasi-projectivity, (co-)homological invariants, Chow groups, (uni-)rationality, ample divisors, etc.)?

Although moduli problems emerged rather early in the history of algebraic geometry, a rigorous formulation of the right concept for them, just in the abstract sense mentioned above, as well as the precise definition of what a moduli scheme should be, has only been given pretty recently. Elaborating the pioneering ideas of A. Grothendieck, and combining them with the framework of invariant theory, in a just as ingenious a manner, D. Mumford established modern moduli theory and geometric invariant theory in the early 1960s. Since the appearance of D. Mumford’s fundamental monograph “Geometric invariant theory” (1965; Zbl 0147.39304), in which he provided the conceptual and methodical principles of algebro-geometric moduli theory, together with an explicit construction of (coarse) moduli schemes for some concrete classes of objects (e.g., smooth algebraic curves, polarized abelian varieties), there has been an explosion of activity in investigating various moduli problems in algebraic geometry. The third, enlarged edition of Mumford’s famous “GIT” [cf.: D. Mumford, J. Fogarty and F. Kirwan, “Geometric invariant theory”, 3rd edition (1993; Zbl 0797.14004)] gives, in seven updated appendices, a good survey on the developments over the past thirty years. – In attempting to show the existence of moduli schemes for concrete moduli problems (via geometric invariant theory), one often has to verify that the quotient of a scheme by a group action is again a scheme, or at least an algebraic space (in the sense of M. Artin). Once having been able to construct a moduli scheme in this way, the question of whether it is (quasi-)projective leads to the problem of constructing ample sheaves on it.

This general approach to constructing moduli schemes, together with a projective embedding, is the central theme of the present book under review. Roughly speaking, the author discusses two subjects of seemingly quite different nature. The first one concerns construction methods for good quotients of quasi-projective schemes modulo group actions (or modulo equivalence relations, respectively), while the second subject of the text incorporates the study of the direct images of higher powers of dualizing sheaves associated with families of complex projective varieties.

The methods developed in the course of the discussion of these two topics are then combined in order to construct quasi-projective moduli spaces for certain types of moduli problems. This approach to a general construction principle for moduli spaces is motivated by (and based upon) the author’s earlier, extensive work on weakly positive sheaves associated with algebra fibre spaces, and their application to the study of Iitaka’s “\(C^+_{n,m}\)-conjecture” [cf.: E. Viehweg, in: Algebraic Varieties and Analytic Varieties, Proc. Symp, Tokyo 1981, Adv. Stud. Pure Math. 1, 329-353 (1983; Zbl 0513.14019)]. The link between the “\(C^+_{n,m}\)-conjecture”, the property of weak positivity for direct image sheaves, and moduli theory for complex manifolds of general type has led the author to investigate this interrelation more closely, and to use “weak positivity” and geometric invariant theory, in a general way, to construct suitable moduli problems and, simultaneously, quasi-projective coarse moduli spaces for them. This has been published in a series of papers which appeared between 1989 and 1991 [cf. H. Esnault and E. Viehweg, in: Algebraic Geometry and analytic Geometry, Proc. Conf., Tokyo 1990, ICM-90 Satell. Conf. Proc., 53-80 (1991; Zbl 0772.14009); E. Viehweg, “Weak positivity and the stability of certain Hilbert points”, I, II, and III, Invent. Math. 96, No. 3, 639-667 (1989; Zbl 0695.14006) and 101, No. 1, 191-223 (1990; Zbl 0721.14007) and No. 3, 521-543 (1990; Zbl 0746.14014); and E. Viehweg, Math. Ann. 289, No. 2, 297-314 (1991; Zbl 0729.14010)].

The present monograph is a systematic, comprehensive and textbook-like elaboration of the author’s work published in these papers. Compared to them, the author has generalized some results (e.g., the stability criteria in chapter 4), simplified many proofs and constructions, and he has given the entire presentation a nearly self-contained character by including (with proofs) most of those results which are perhaps well-known to specialists but barely documented in the standard texts on algebraic geometry.

Chapter 1 provides therefore the basic material from moduli theory: moduli problems, moduli functors, coarse and fine moduli schemes, Grothendieck’s construction of Hilbert schemes, and Hilbert schemes of (canonically) polarized schemes. – Chapter 2 deals with numerically effective and weakly positive sheaves, including the author’s covering constructions and generalizations of the Fujita-Kawamata positivity theorem. The basic definitions and results from Mumford’s geometric invariant theory are compiled in chapter 3, while chapter 4 turns to the author’s approach to constructing stable points in Hilbert schemes by means of weakly positive sheaves and the ampleness criteria of Viehweg and Kollár. Chapter 5 interrupts the program of constructing moduli schemes of canonically polarized manifolds a little bit. Auxiliary results on locally free sheaves and divisors, including an unpublished theorem of O. Gabber on extensions of certain locally free sheaves to compactifications, a discussion of singularities of divisors (in flat families), and some vanishing theorems, form the heart of this more technical chapter. It is, however, very important in the sequel, when moduli schemes also for non-canonically polarized manifolds are to be constructed. – Chapter 6 presents the author’s methods and results on weakly positive direct image sheaves, as they can be found in his original papers. Using O. Gabber’s extension theorem, the author presents the material in a very elegant, simplified manner. This is then used, in chapter 7, to construct moduli schemes and, more generally, geometric quotients. The ampleness criteria from chapter 4 even provide quasi-projective geometric quotients. In order to establish the existence of moduli schemes for certain classes of singular varieties, the author discusses (at the end of chapter 7) the conditions for that, in the case of larger classes of moduli functors. – Chapter 8 turns then to moduli functors of normal varieties with canonical singularities. It is shown that, under additional assumptions (boundedness, local closedness, separatedness) for such functors, the method of constructing quasi-projective moduli schemes (developed so far) carries over this case, too, including even non-canonically polarized varieties. Chapter 9, the concluding chapter, discusses another approach of constructing moduli schemes, namely the one via M. Artin’s theory of algebraic spaces (or, in the complex case, via the theory of Moishezon spaces). After a brief introduction to this framework, the author explains how to construct quotients by equivalence relations in the category of algebraic spaces, and to prove then algebraicity and quasi-projectivity by methods developed by himself and by J. Kollár [J. Differ. Geom. 32, No. 1, 235-268 (1990; Zbl 0684.14002)]. – The author’s general construction principle for quasi-projective moduli schemes re-establishes most of the known existence theorems for curves, abelian varieties, surfaces of general type, \(K\)-3 surfaces, etc., and that in a unified way. However, in some cases, the ample line bundles that he uses to prove quasi-projectivity are less nice and explicit as in the constructions given before.

Altogether, the book under review is an outstanding research monograph on moduli theory. It presents the subject and its very recent achievements in a most general, systematic and comprehensive manner. As detailed and self-contained as it is, the text may be regarded as a worthy, contemporary successor of D. Mumford’s celebrated “Geometric invariant theory” (loc. cit.).

Reviewer: W.Kleinert (Berlin)

### MSC:

14D20 | Algebraic moduli problems, moduli of vector bundles |

14D22 | Fine and coarse moduli spaces |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |