For a positive integer g, we let

ℍg=τ∈ℂ(g,g)| τ=tτ, Imτ>0

be the Siegel upper half plane of degree g

and let

Sp(2g,ℝ)={M∈ ℝ(2g,2g) | t​MJgM=Jg}

be the symplectic group of degree g, where F(k,l) denotes the set of all k×l matrices with entries in a commutative ring F for two positive integers k and l, t​M denotes the transposed matrix of a matrix M and

Jg=0Ig−Ig0.

Then Sp(2g,ℝ) acts on ℍg transitively by

(1.1)M⋅τ=(Aτ+B)(Cτ+D)−1,

where M=ABCD∈Sp(2g,ℝ) and Ω∈ℍn. Let

Γg=Sp(2g,ℤ)=ABCD∈Sp(2g,ℝ)| A,B,C,D integral

be the Siegel modular group of degree g. This group acts on ℍg properly discontinuously.

Let Ag:=Γg\ℍg be the Siegel modular variety of degree g, that is, the moduli space of g-dimensional principally polarized abelian varieties, and let Mg be the the moduli space of projective curves of genus g. Then according to Torelli's theorem, the Jacobi mapping

(1.2)Tg:Mg→Ag

defined by

C↦J(C):=the Jacobian of C

is injective. The Jacobian locus Jg:=Tg(Mg) is a (3g−3)-dimensional subvariety of Ag

The Schottky problem is to characterize the Jacobian locus or its closure J¯g in Ag. At first this problem had been investigated from the analytical point of view : to find explicit equations of J_g (or J¯g) in Ag defined by Siegel modular forms on ℍg, for example, polynomials in the theta constant θϵδ(τ,0) (see Definition (2.4)) and their derivatives. The first result in this direction was due to Friedrich Schottky [125] who gave the simple and beautiful equation satisfied by the theta constants of Jacobians of dimension 4. Much later the fact that this equation characterizes the Jacobian locus J₄ was proved by J. Igusa [73] (see also E. Freitag [47] and Harris-Hulek [68]). Past decades there has been some progress on the characterization of Jacobians by some mathematicians. Arbarello and De Concini [6] gave a set of such equations defining J¯g. The Novikov conjecture which states that a theta function satisfying the Kadomtsev-Petviasvili (briefly, K-P) differential equation is the theta function of a Jacobian was proved by T. Shiota [129]. Later the proof of the above Novikov conjecture was simplified by Arbarello and De Concini [7]. Bert van Geeman [53] showed that J¯g is an irreducible component of the subvariety of AgSat defined by certain equations. Here AgSat is the Satake compactification of Ag. I. Krichever [80] proved that the existence of one trisecant line of the associated Kummer variety characterizes Jacobian varieties among principally polarized abelian varieties.

S.-T. Yau and Y. Zhang [177] obtained the interesting results about asymptotic behaviors of logarithmical canonical line bundles on toroidal compactifications of the Siegel modular varieties. Working on log-concavity of multiplicities in representation theory, A. Okounkov [106, 107] showed that one could associate a convex body to a linear system on a projective variety, and use convex geometry to study such linear systems. Thereafter R. Lazarsfeld and M. Mustată [81] developed the theory of Okounkov convex bodies associated to linear series systematically. E. Freitag [45] introduced the concept of stable modular forms to investigate the geometry of the Siegel modular varieties. In 2014, using stable modular forms, G. Codogni and N. I. Shepherd-Barron [24] showed there is no stable Schottky-Siegel forms. We recall that Schottky-Siegel forms are scalar-valued Siegel modular forms vanishing on the Jacobian locus. Recently G. Codogni [23] found the ideal of stable equations of the hyperelliptic locus. About twenty years ago the author [148, 158] introduced the notion of stable Jacobi forms to try to study the geometry of the universal abelian varieties. In this paper, we discuss the relations among logarithmical line bundles on toroidal compactifications, the André-Oort conjecture, Okounkov convex bodies, Coleman's conjecture, Siegel-Jacobi spaces, stable Schottky-Siegel forms, stable Schottky-Jacobi forms and the Schottky problem.

This article is organized as follows. In Section 2, we briefly survey some known approaches to the Schottky problem and some results so far obtained concerning the characterization of Jacobians. In Section 3, we briefly describe the results of Yau and Zhang concerning the behaviors of logarithmical canonical line bundles on toroidal compactifications of the Siegel modular varieties. In Section 4, we review some recent progress on the André-Oort conjecture. In Section 5, we review the theory of Okounkov convex bodies associated to linear series (cf. [20, 81]). In Section 6, we discuss the relations among logarithmical line bundles on toroidal compactifications, the André-Oort conjecture, Okounkov convex bodies, Coleman's conjecture and the Schottky problem. In the final section we give some remarks and propose some open problems about the relations among the Schottky problem, the André-Oort conjecture, Okounkov convex bodies, stable Schottky-Siegel forms, stable Schottky-Jacobi forms and the geometry of the Siegel-Jacobi space. We define the notion of stable Schottky-Jacobi forms and the concept of stable Jacobi equations for the universal hyperelliptic locus. In Appendix A, we survey some known results about subvarieties of the Siegel modular variety. In Appendix B, we review recent results concerning an extension of the Torelli map to a toroidal compactification of the Siegel modular variety. In Appendix C, we describe why the study of singular modular forms is closely related to that of the geometry of the Siegel modular variety. In Appendix D, we briefly talk about singular Jacobi forms. Finally in Appendix E, we review the concept of stable Jacobi forms introduced by the author and relate the study of stable Jacobi forms to that of the geometry of the universal abelian variety.

Finally the author would like to mention that he tried to write this article in another new perspective concerning the Schottky problem different from that of other mathematicians. The list of references in this article is by no means complete though we have strived to give as many references as possible. Any inadvertent omissions of references related to the contents in this paper will be the author's fault.

Notations: We denote by ℚ,ℝ and ℂ the field of rational numbers, the field of real numbers and the field of complex numbers respectively. We denote by ℤ and ℤ+ the ring of integers and the set of all positive integers respectively. ℝ+ denotes the set of all positive real numbers. ℤ+ and ℝ+ denote the set of all nonnegative integers and the set of all nonnegative real numbers respectively. The symbol ":=" means that the expression on the right is the definition of that on the left. For two positive integers k and l, F(k,l) denotes the set of all k×l matrices with entries in a commutative ring F. For a square matrix A∈F(k,k) of degree k, tr(A) denotes the trace of A. For any M∈F(k,l),t​M denotes the transpose of a matrix M. I_n denotes the identity matrix of degree n. For A∈F(k,l) and B∈F(k,k), we set B[A]=tABA. For a complex matrix A,A¯ denotes the complex conjugate of A. For A∈ℂ(k,l) and B∈ℂ(k,k), we use the abbreviation B{A}=tA¯BA. For a number field F, we denote by AF,f the ring of finite adéles of F.

Before we survey some approaches to the Schottky problem, we provide some notations and definitions. Most of the materials in this section can be found in [60]. We refer to [13, 31, 41, 60, 101, 117] for more details and discussions on the Schottky problem.

In this section, we let g be a fixed positive integer. For a positive integer ℓ, we define the principal level ℓ subgroup

Γg(l):=γ∈Sp(2g,ℤ)| γ≡I2g (mod l).

and define the theta level ℓ subgroup

Γg(l,2l):=ABCD∈Γg(l)| diag(t​AB)≡diag(tCD)≡0 (mod l).

We let

Ag(l):=Γg(l)\ℍg and Ag(l,2l):=Γg(l,2l)\ℍg.

Definition 2.1. ([72, pp.,49-50], [100, p.,123], [156, p.,862] or [167, p.,127]) Let ℓ a positive integer. For any ϵ and δ in 1lℤg/ℤg, we define the theta function with characteristics ϵ and δ by

(2.1)θϵδ(τ,z):=∑ N∈ℤgeπi{(N+ϵ) τ t (N+ϵ) + 2 (N+ϵ) t (z+δ)}, (τ,z)∈ℍg×ℂg.

The Riemann theta function θ(τ,z) is defined to be

(2.2)θ(τ,z):=θ00(τ,z), (τ,z)∈ℍg×ℂg.

For each τ∈ℍg, we have the transformation behavior

(2.3)θ(τ,z+aτ+b):=e−πi(aτ t​a + 2 at​z)θ(τ,z) for all a,b∈ℤg.

The function

(2.4)θϵδ(τ):=θϵδ(τ,0), τ∈ℍg

is called the theta constant of order ℓ. It is known that the theta constants θϵδ(τ) of order ℓ are Siegel modular forms of weight 12 for Γg(l,2l),[100, p.,200].

For a fixed τ∈ℍg, we let Λτ:=ℤgτ+ℤg be the lattice in ℂg. According to the formula (2.3), the zero locus z∈ℂg| θ(τ,z)=0 is invariant under the action of the lattice Λτ on ℂg, and thus descends to a well-defined subvariety Θτ⊂Aτ:=ℂg/Λτ. In fact Aτ is a principally polarized abelian variety with ample divisor Θτ.

Definition 2.2. For ϵ∈12ℤg/ℤg the theta function of the second order with characteristic ϵ is defined to be

(2.5)Θ[ϵ](τ,z):=θ2ϵ0(2τ,2z), (τ,z)∈ℍg×ℂg.

We define the theta constant of the second order to be

(2.6)Θ[ϵ](τ):=θ2ϵ0(2τ,0)=Θ[ϵ](τ,0), τ∈ℍg.

Then we see that Θ[ϵ](τ) is a Siegel modular form of weight 12 for Γg(2,4).

We have the following results.

Theorem 2.1. (Riemann's bilinear addition formula),[72, p.,139]

(2.7)θϵδ(τ,z)2=∑ σ∈1 2 ℤg /ℤg (−1)4 t​σδΘ[σ+ϵ](τ,0)⋅Θ[σ](τ,z).

Theorem 2.2. For l≥2, the map

(2.8)Φl:Ag(2l,4l)→ℙN(ℂ), N:=l2g−1

defined by

Φl(τ):=θϵδ(τ)| ϵ,δ∈1lℤg/ℤg

is an embedding.

Proof. See Igusa [72] for l=4n2, and Salvati Manni [120] for l≥2.

Remark 2.1. We consider the theta map

(2.9)Th:Ag(2,4)→ℙ2g−1(ℂ)

defined by

Th(τ):=Θ[ϵ](τ)| ϵ∈12ℤg/ℤg.

We observe that according to Theorem 2.1, Φ2(τ) can be recovered uniquely up to signs from Θ(τ). Since Φ2 is injective on Ag(4,8), the theta map Th is finite-to-one on Ag(2,4). In fact, it is known that the theta map Th is generically injective, and it is conjectured that Th is an embedding.

Now we briefly survey some approaches to the Schottky problem. As mentioned before, most of the following materials in this section comes from a good survey paper [60].

(A) Classical Approach

For τ∈ℍg and a positive integer l∈ℤg,

Aτ[l]:=τϵ+δ∈Aτ| ϵ,δ∈1lℤg/ℤg

denotes the subgroup of A_τ consisting of torsion points of order ℓ. For m=τϵ+δ∈Aτ[l], we briefly write

(2.10)θm(τ,z):=θϵδ(τ,z).

We define the Igusa modular form to be

(2.11)Fg(τ):=2g∑ m∈Aτ[2]θm16(τ)−∑m∈Aτ [2] θm8(τ)2, τ∈ℍg.

It was proved that Fg(τ) is a Siegel modular form of weight 8 for the Siegel modular group Γ_g such that when rewritten in terms of theta constants of the second order using Theorem 2.1,

(Fg−1)    Fg≡0    for  g=1,2;

(Fg−2)  F3 is the defining equation for Th(J3(2,4))¯=Th(A3(2,4))¯⊂ℙ7(ℂ);

(Fg−3)  F4 is the defining equation for Th(J4(2,4))¯⊂Th(A4(2,4))¯⊂ℙ15(ℂ).

For more detail, we refer to [47, 73, 125] for the case g=4 and refer to [117] for the case g=5. For g≥ 5, no similar solution is known or has been proposed.

Theorem 2.3. If g≥ 5, then F_g does not vanish identically on J_g. In fact, the zero locus of F₅ on J₅ is the locus of trigonal curves.

The above theorem was proved by Grushevsky and Salvati Manni [64].

(B) The Schottky-Jung Approach

Definition 2.3. For an étale connected double cover C˜→C of a curve C∈Mg (such a curve is given by a two-torsion point η(=0)∈J(C)[2]) we define the Prym variety to

Prym(C˜→C):=Prym(C,η):=Ker0(J(C˜)→J(C))∈Ag−1,

where Ker0 denotes the connected component of 0 in the kernel and the map J(C˜)→J(C) is the norm map corresponding to the cover C˜→C. We denote by Pg−1⊂Ag−1 the locus of Pryms of all étale double covers of curves in Mg.

The problem of describing Pg−1 is called the Prym-Schottky problem.

Remark 2.2. The restriction of the principal polarization ΘJ(C˜) to the Prym gives twice the principal polarization. However this polarization admits a canonical square root, which thus gives a natural principal polarization on the Prym.

Theorem 2.4. (Schottky-Jung proportionality) Let τ be the period matrix of a curve C of genus g and let τ* be the period matrix of the Prym for 00⋯010⋯0. Then for any ϵ,δ∈12ℤg−1/ℤg−1 the theta constants of J(C) and of the Prym are related by

(2.12)θϵδ(τ*)2=Cθ0ϵ0δ(τ)⋅θ0ϵ1δ(τ).

Here the constant C is independent of ϵ,δ.

Proof. See Schottky-Jung [126] and also Farkas [41] for a rigorous proof.

Definition 2.4. (The Schottky-Jung locus [60]). Let I_g-1 be the defining ideal for the image Th(Ag−1(2,4))¯⊂ℙ2g−1−1 (see Remark 2.1). For any equation F∈Ig−1, we let Fη be the polynomial equation on ℙ2g−1 obtained by using the Schottky-Jung proportionality to substitute an appropriate polynomial of degree 2 in terms of theta constants of τ for the square of any theta constant of τ*. Let Sgη be the ideal obtained from Ig−1 in this way. We define the big Schottky-Jung locus Sgη(2,4)⊂Ag(2,4) to be the zero locus of Sgη. It is not known that Ig⊂Sgη and thus we define Sgη(2,4) within Ag(2,4), and not as a subvariety of the projective space ℙ2g−1. We now define the small Schottky-Jung locus to be

(2.13)Sg(2,4):=∩ηSg η(2,4),

where η runs over the set 12ℤ2g/ℤ2g\{0}. We note that the action of Γg permutes the different η and the ideals Sgη. Therefore the ideal defining Sg(2,4) is Γg-invariant, and the locus Sg(2,4) is a preimage of some Sg⊂Ag under the level cover.

Theorem 2.5. (a) The Jacobian locus J_g is an irreducible component of the small Schottky-Jung locus Sg.

(b) Jg(2,4) is an irreducible component of the big Schottky-Jung locus Sgη(2,4) for any η.

Proof. The statement (a) was proved by van Geeman [53] and the statement (b) was proved by Donagi [31].

Donagi [32] conjectured the following.

Conjecture 2.1. The small Schottky-Jung locus is equal to the Jacobian locus, that is, Sg=Jg.

(C) The Andreotti-Mayer Approach

We let SingΘ be the singularity set of the theta divisor Θ for a principally polarized abelian variety (A,Θ).

Theorem 2.6. For a non-hyperelliptic curve C of genus g, dim(SingΘJ(C))=g−4, and for a hyperelliptic curve C, dim(SingΘJ(C))=g−3. For a generic principally polarized abelian variety, the theta divisor is smooth.

Proof. The proof was given by Andreotti and Mayer [5].

Definition 2.5. We define the k-th Andreotti-Mayler locus to be

Nk, g:={(A,Θ)∈Ag| dimSingΘ≥k}.

Theorem 2.7. Ng−2,g=Agdec. Here

Agdec:=∪ k=1 g−1Ak×Ag−k⊂Ag

denotes the locus of decomposable ppavs (product of lower-dimensional ppavs) of dimension g.

Proof. The proof was given by Ein and Lazasfeld [38].

Theorem 2.8. J_g is an irreducible component of Ng−4,g, and the locus of hyperelliptic Jacobians Hypg is an irreducible component of Ng−3,g.

Proof. The proof was given by Andreotti and Mayer [5].

Theorem 2.9. The Prym locus Pg is an irreducible component of Ng−6,g.

Proof. The proof was given by Debarre [26].

Theorem 2.10. The locus of Jacobians of curves of genus 4 with a vanishing theta-null is equal to the locus of 4-dimensional principally polarized abelian varieties for which the double point singularity of the theta divisor is not ordinary (i.e., the tangent cone does not have maximal rank).

Proof. See Grushevsky-Salvati Manni [63] and Smith-Varley [132].

Problem. Can it happen that Nk,g=Nk+1,g for some k,g?

(D) The Approach via the K-P Equation

In his study of solutins of nonlinear equations of Korteveg de Vrie type, I. Krichever [79] proved the following fact:

Theorem 2.11. Let τ be the period matrix of a curve C of genus g and let θ(z) (cf. (2.2)) be the Riemann theta function of the Jacobian J(C). Then there exist three vectors W1,W2,W3 in ℂg with W1=0 such that, for every z∈ℂg, the function

(2.14)u(x,y,z;t):=∂2 ∂x2logθ(xW1+yW2+tW3+z)

satisfies the so-called Kadomstev-Petriashvili\ equation (briefly the K-P equation)

(2.15)3uyy=ut −3uux −2uxxx x.

S.P. Novikov conjectured that τ∈ℍg is the period matrix of a curve if and only if the Riemann theta function corresponding to τ∈ℍg satisfies the K-P equation in the sense we just explained in Theorem 2.11. Shiota [129] proved that the Novikov conjecture is true, following the work of Mulase [96] and Mumford [98].

Arbarello and De Concini [7] gave another proof of the Novikov conjecture.

(E) The Approach via Geometry of the Kummer Variety

Definition 2.6. The map is the embedding given by

(2.16)Kum:Aτ/±1→ℙ2g−1(ℂ),  Kum(z)=Θ[ϵ](τ,z)| ϵ∈12ℤg/ℤg.

We call the image of Kum the Kummer variety. Note that the involution ±1 has 22g fixed points on Aτ which are precisely Aτ[2], and thus the Kummer variety singular at their images in ℙ2g−1(ℂ).

Theorem 2.12. For any points p1,p2,p3 of a curve of genus g, the following three points on the Kummer variety are collinear:

(2.17)Kum(p+p1−p2−p3), Kum(p+p2−p1−p3), Kum(p+p3−p1−p2).

Proof. See Fay [42] and Gunning [65].

Theorem 2.13. For any curve C∈Mg, for any 1≤k≤g and for any p1,⋯,pk+2,q1,⋯,qk∈C the k+2 points of the Kummer variety

(2.18)Kum2pj+∑ i=1kqi−∑ i=1 k=2pi, j=1,⋯,k+2

are linearly dependent.

Proof. See Gunning [66].

I. Krichever [80] gave a complete proof of a conjecture of Welters concerning a condition for an indecomposable principally polarized abelian variety to be the Jacobian of a curve:

Theorem 2.14. Let Agind:=Ag\Agdec be the locus of indecomposable ppavs of dimension g. For a ppav A∈Agind, if Kum(A)⊂ℙ2g−1 has one of the following

(W1) a trisecant line

(W2) a line tangent to it at one point, and intersecting it another point

(this is a semi-degenerate trisecant, when two points of secancy coincides)

(W3) a flex line (this is a most degenerate trisecant when all three points of secancy coincide)

such that none of the points of intersection of this line with the Kummer variety are A[2] (where Kum(A) is singular), then A∈Jg.

For the Prym-Schottky problem, it will be natural whether the Kummer varieties of Pryms have any special geometric properties. Indeed, Beauville-Debarre [14] and Fay [43] obtained the following.

Theorem 2.15. Let C∈Mg. For any p,p1,p2,p3∈C˜→Prym(C˜→C) on the Abel-Prym curve the following four points of the Kummer variety

Kum(p+p1+p2+p3), Kum(p+p1−p2−p3),Kum(p+p2−p1−p3), Kum(p+p3−p1−p2)

lie on a 2-plane in ℙ2g−1(ℂ).

A suitable analog of the trisecant conjecture was found for Pryms using ideas of integrable systems by Grushevsky and Krichever [62]. They proved the following.

Theorem 2.16. If for some A∈Agind and some p,p1,p2,p3∈A the quadrisecant condition in Theorem 2.15 holds, and

moreover there exists another quadrisecant given by Theorem 2.15 with p replaced by -p, then A∈Pg˜.

(F) The Approach via the Γ00 Conjecture

Definition 2.7. Let (A,Θ)∈Ag. The linear system Γ00⊂|2Θ| is defined to consist of those sections that vanish to order at least 4 at the origin:

(2.19)Γ00:={f∈H0(A,2Θ)| mult0f≥4}.

We define the base locus

FA:={x∈A| s(x)=0 for all s∈Γ00}.

Theorem 2.17. For any g≥ 5 and any C∈Mg, we have on the Jacobian J(C) of C the equality

FJ(C)=C−C={x−y∈J(C)| x,y∈C}.

Proof. The above theorem was proved by Welters [143] set theoretically and also by Izadi scheme-theoretically. Originally Theorem 2.17 was conjectured by van Geeman and van der Geer [54].

van Geeman and van der Geer [54] conjectured the following.

Γ00 Conjecture. Let (A,Θ)∈Agind. If FA≠0, then A∈Jg.

Definition 2.8. Let (A,Θ)∈Ag. For any curve Γ on A and any point x∈ A, we define

ε(A,Γ,x):=Θ.ΓmultxΓ,  ε(A,x):=infΓ∋xε(A,Γ,x).

We define the Seshadri constant of (A,Θ) by

ε(A):=ε(A,Θ):=infx∈Aε(A,x).

Theorem 2.18. If the Γ00 conjecture holds, hyperelliptic Jacobians are characterized by the value of their Seshadri constants.

Proof. See O. Debarre [28].

Theorem 2.19. If some A∈Agind, the linear dependence

(2.20)Θ[ϵ](τ,z)=cΘ[ϵ](τ,0)+∑ 1≤a≤b≤gcab∂Θ[ϵ](τ,0)∂τab

for some c,cab∈ℂ (1≤a≤b≤g) and for all ϵ∈12ℤg/ℤg holds with rank(cab)=1, then A∈Jg.

Proof. See S. Grushevsky [59].

(G) Subvarieties of a ppav: Minimal Cohomology Classes

The existence of some special subvarieties of a ppav (A,Θ)∈Ag gives a criterion that A is the Jacobian of a curve. We start by observing that for the Jacobian J(C) of a curve C∈Mg we can map the symmetric product Symd(C)(1≤d<g) to J(C)=Picg−1(C) by fixing a divisor D∈Picg−1−d(C) and mapping

(2.21)Φ(d):Symd(C)→J(C),  (p1,⋯,pd)↦D+∑ i=1 dpi.

The image Wd(C) of the map Φ(d) is independent of D up to translation, and we can compute its cohomology class

Wd(C)=[Θ]d(g−d)!∈H2g−2d(J(C)),

where [Θ] is the cohomology class of the polarization of J(C). One can show that the cohomology class is indivisible in cohomology with ℤ-coefficients, and we thus call this class minimal. We note thatW1(C)≃C. These subvarieties Wd(C) (1≤d<g) are very special.

We have the following criterion.

Theorem 2.20. A ppav (A,Θ)∈Ag is a Jacobian if and only if there exists a curve C⊂A with [C]=[Θ]g−1(g−1)!∈H2g−2(J(C)) in which case (A,Θ)=J(C).

Proof. See Matsusaka [92] and Ran [116].

Debarre [27] proved that J_g is an irreducible component of the locus of ppavs for which there is a subvariety of the minimal cohomology class. He conjectured the following.

Conjecture 2.2. If a ppav (A,Θ)∈Ag has a d-dimensional subvariety of minimal class, then it is either the Jacobian of a curve or a five dimensional intermediate Jacobian of a cubic threefold.

This approach to the Schottky problem gives a complete geometric solution to the weaker version of the problem: determining whether a given ppav is the Jacobian of a given curve.

In this section, we review the interesting results obtained by S.-T. Yau and Y. Zhang [177] concerning the asymptotic behaviors of the logarithmical canonical line bundle on a toroidal compactification of the Siegel modular variety.

Let Γ be a neat arithmetic subgroup of Γg. Let Ag,Γ:=Γ\ℍg and A¯g,Γ be the toroidal compactification of Ag,Γ constructed by a GL(g,ℤ)-admissible family of polyhedral decompositions ΣF0 of the cones. Here F0 denotes the standard minimal cusps of ℍg. A¯g,Γ is an algebraic space, but a projective variety in general. Y.-S. Tai proved that if ΣtorΓ is projective (see [9, Chapter IV, Corollary 2.3, p.,200]), then A¯g,Γ is a projective variety. It is known that A¯g,Γ is the unique Hausdorff analytic variety containing Ag,Γ as an open dense subset (cf. [9]).

Assume the boundary divisor D∞,Γ:=A¯g,Γ\Ag,Γ is simple normal crossing. We put N=g(g+1)/2. For each irreducible component D_i of D∞,Γ=∪jD j, let s_i a global section of the line bundle [D_i] defining D_i. Let σmax be an arbitrary top-dimensional cone in ΣF0 and renumber all components D_i's of D∞,Γ such that D1,⋯,DN correspond to the edges of σmax with marking order. Yau and Zhang [177, Theorem 3.2] showed that the volume form Φg,Γ on Ag,Γ may be written by

Φg,Γ=2N−gVolΓ(σmax)2dVg∏ j=1N‖sj ‖2Fσmaxg+1(log‖s1‖1,⋯,log‖sN‖N),

where dVg is a continuous volume form on a partial compactification Uσmax of Ag,Γ with Ag,Γ⊂Uσmax⊂A¯g,Γ, and each ‖⋅‖j is a suitable Hermitian metric of the line bundle [D_j] on A¯g,Γ (1≤j≤N) and Fσmax∈ℤ[x1,⋯,xN] is a homogeneous polynomial of degree g. Moreover the coefficients of Fσmax depends only on both Γ and σmax with marking order of edges. Using the above volume form formula they showed that the unique invariant Kähler-Einstein metric on Ag,Γ endows some restraint combinatorial conditions for all smooth toroidal compactifications of Ag,Γ.

Let E1,⋯,Ed be any different irreducible components of the boundary divisor D∞,Γ such that ∩ k=1dEk≠∅. Let Kg,Γ be the canonical line bundle on A¯g,Γ. Yau and Zhang [177] also proved the following facts (a) and (b):

(a) Let i1,⋯,id∈ℤ+. If d≥g−1 and N−∑ k=1dik>2 (or if d≥g and N−∑ k=1dik=1),

then we have

Kg,Γ+D∞,ΓN−∑ k=1dik⋅E1i1⋯Edid=0.

(b) Kg,Γ+D∞,Γ is not ample on A¯g,Γ.

They also showed that if d<g−1, then the intersection number

Kg,Γ+D∞,ΓN−d⋅E1⋯Ed

can be expressed explicitly using the above volume form formula. The proofs of (a) and (b) can be found in [177, Theorem 4.15].

In this section we review recent progress on the André-Oort conjecture quite briefly.

Definition 4.1. Let (G,X) be a Shimura datum and let K be a compact open subgroup of G(Af). We let

ShK(G,X):=G(ℚ)\X×G(Af)/K

be the Shimura variety associated to (G,X). An algebraic subvariety Z of the Shimura variety Sh_K(G,X) is said to be weakly special if there exist a Shimura sub-datum (H,XH) of (G,X), and a decomposition

(Had,XHad)=(H1,X1)×(H2,X2)

and y2∈X2 such that Z is the image of X1×{y2} in ShK(G,X). Here (Had,XHad) denotes the adjoint Shimura datum associated to (G,X) and (Hi,Xi) (i=1,2) are Shimura data. In this definition, a weakly special subvariety is said to be special if it contains a special point and y₂ is special.

André [4] and Oort [108] made conjectures analogous to the Manin-Mumford conjecture where the ambient variety is a Shimura variety (the latter partially motivated by a conjecture of Coleman [25]). A combination of these has become known as the André-Oort conjecture (briefly the A-O conjecture).

A-O Conjecture. Let S be a Shimura variety and let Σ be a set of special points in S. Then every irreducible component of the Zariski closure of Σ is a special subvariety.

Definition 4.2. [111, 112] A pre-structure is a sequence Σ=(Σn:n≥1) where each Σn is a collection of subsets of ℝn. A pre-structure Σ is called a structure over the real field if, for all n,m≥1 with m≤n, the following conditions are satisfied:

(1) Σn is a Boolean algebra (under the usual set-theoretic operations);

(2) Σn contains every semi-algebraic subset of ℝn;

(3) if A∈Σm and B∈Σn, then A×B∈Σm+n;

(4) if n≥ m and A∈ Σ_n, then πn,m(A)∈Σm, where πn,m:ℝn→ℝm is a coordinate

projection on the first m coordinates.

If Σ is a structure, and, in addition,

(5) the boundary of every set in Σ1 is finite,

then Σ is called an o-minimal structure over the real field.

If Σ is a structure and Z⊂ℝn, then we say that Z is definable in Σ if Z∈Σn. A function f:A→B is definable in a structure Σ if its graph is definable, in which case the domain A of f and image f(A) are also definable by the definition. If A,⋯,f,⋯ are sets or functions, then we denote by ℝA,⋯,f,⋯ the smallest structure containing A,⋯,f,⋯. By a definable family of sets we mean a definable subset Z⊂ℝn×ℝm which we view as a family of fibres Zy⊂ℝn as y varies over the projection of Z onto ℝm which is definable, along with all the fibres Z_y. A family of functions is said to be definable if the family of their graphs is. A definable set usually means a definable set in some o-minimal structure over the real field.

Remark 4.1. The notion of a o-minimal structure grew out of work van den Dries [33, 34] on Tarski's problem concerning the decidability of the real ordered field with the exponential function, and was studied in the more general context of linearly ordered structures by Pillay and Steinhorn,[115], to whom the term "o-minimal" ("order-minimal") is due.

In 2011 Pila gave a unconditional proof of the A-O conjecture for arbitrary products of modular curves using the theory of o-minimality.

Theorem 4.1. Let

X=Y1×⋯×Yn×E1×⋯×Em×Gml,

where n,m,l≥0, Yi=Γ(i)\ℍ1(1≤i≤n) are modular curves corresponding to congruence subgroups Γ(i) of SL(2,ℤ) and Ej(1≤j≤m) are elliptic curves defined over ℚ¯ and Gm is the multiplicative group. Suppose V is a subset of X. Then V contains only a finite number of maximal special subvarieties.

Proof. See Pila [111, Theorem 1.1].

In 2013 Peterzil and Starchenko proved the following theorem using the theory of o-minimality.

Theorem 4.2. The restriction of the uniformizing map $π:ℍg→Ag to the classical fundamental domain for the Siegel modular group Sp(2g,ℤ) is definable.

Proof. See Peterzil and Starchenko [109, 110].

In 2014 Pila and Tsimerman gave a conditional proof of the A-O conjecture for the Siegel modular variety Ag.

Theorem 4.3. If g≤6, then the A-O conjecture holds for Ag. If g≥ 7, the A-O conjecture holds for Ag under the assumption of the Generalized Riemann Hypothesis (GRH) for CM fields.

Proof. See Pila-Tsimerman [113, 114].

Quite recently using Galois-theoretic techniques and geometric properties of Hecke correpondences, Klingler and Yafaev proved the A-O conjecture for a general Shimura variety, and independently using Galois-theoretic and ergodic techniques Ullmo and Yafaev proved the A-O conjecture for a general Shimura variety, under the assumption of the GRH for CM fields or another suitable assumption. The explicit statement is given as follows.

Theorem 4.4.Let (G,X) be a Shimura datum and K a compact open subgroup of G(Af). Let Σ be a set of special points in ShK(G,X). We make one of the two following assumptions:

(1) Assume the GRH for CM fields.

(2) Assume that there exists a faithful representation G↪GLn such that with respect to this representation, the Mumford-Tate group MT(s) lie in one GLn(ℚ)-conjugacy class as s ranges through Σ. Then every irreducible component of Σ in Sh_K(G,X) is a special subvariety.

Proof. See Klingler-Yafaev [76] and Ullmo-Yafaev [136].

Remark 4.2. We refer to [112] for the theory of o-minimality and the A-O conjecture. We also refer to [52] for the A-O conjecture for mixed Shimura varieties.

In this section, we briefly review the theory of Okounkov convex bodies associated to pseudoeffective divisors on a smooth projective variety. For more details of this theory, we refer to [20, 81].

Let X be a smooth projective variety of dimension d. We fix an admissible flag Y• on X

Y•: X=Y0⊃Y1⊃Y2⊃⋯⊃Yd−1⊃Yd={x},

where each Y_k is a subvariety of X of codimension k which is nonsingular at x. We let ℤ+ denote the set of all non-negative integers. We first assume that D is a big Cartier divisor on X. For a section s∈H0(X,OX(D))\{0}, we define the function

ν(s)=νY•(s):=(ν1(s),⋯,νd(s))∈ℤ+d

as follows:

First we set ν1=ν1(s):=ordY1(s). Using a local equation f₁ for Y₁ in X, we define naturally a section

s1′=s⊗f1−ν 1∈H0(X,OX(D−ν1Y1))

that does not vanish along Y₁, its restriction s1′|Y 1 defines a nonzero section

s1:=s 1′|Y1∈H0(Y1,OY1(D−ν1Y1)).

We now take

ν2(s):=ordY2(s1).

and continue in this manner to define the remaining νi(s).

Next we define

vect(|D|)=ImνY•:(H0(X,OX(D))−{0})→ℤd

be the set of valuation vectors of non-zero sections of OX(D). Then we finally set

(5.1)Δ(D):=ΔY•(D)=closed convex hull∪ m≥11m⋅vect(|mD|).

Therefore Δ(D) is a convex body in ℝd that is called the Okounkov body of D with respect to the fixed flag Y•. We refer to [81, § 1.2] for some properties and examples of Δ(D).

We recall that a {\sf graded linear series} W•(D)={Wm(D)}m≥0 associated to D consists of subspaces

Wm:=Wm(D)⊆H0(X,OX(mD)),  W0=ℂ

satisfying the inclusion

Wk⋅Wl⊆Wk+l  for all k,l≥0.

Here the product on the left denotes the image of Wk⊗Wl under the multiplication map H0(X,OX(kD))⊗H0(X,OX(lD))→H0(X,OX((k+l)D)).

Definition 5.1. ([81, Definition 1.16]) Let W• be a graded linear series on X belonging to a divisor D. The graded semigroup of W• is defined to be

Γ(W•):=ΓY•(W•)=(νY• (s),m)| 0≠s∈Wm, m≥0⊆ℤ+d×ℤ+⊆ℤd+1.

Under the above notations, we associate the convex body ΔY•(W•) of a graded linear series W• with respect to Y• on X as follows:

(5.2)ΔY•(W•):=∑Γ(W•)∩ℝ+d×{1},

where ℝ+ denotes the set of all non-negative real numbers and ∑Γ(W•) denotes the closure of the convex cone in ℝ+d×ℝ+ spanned by Γ(W•). ΔY•(W•) is called the Okounkov body of W• with respect to Y•. If W• is a complete graded linear series, that is, Wm=H0(X,O(mD)) for each m, then we define

(5.3)ΔY•(D):=ΔY•(W•).

Remark 5.1. ΔY•(D) depends on the choice of an admissible flag Y•. By the homogeneity of ΔY•(D) (see [81, Proposition 4.13]), we can extend the construction of ΔY•(D) to ℚ-divisors D and even to ℝ-divisors using the continuity of ΔY•(D).

Definition 5.2. ([81, Definition 2.5 and 2.9])

(I) We say that a graded linear series W• satisfies Condition (B)

if Wm≠0 for all m≫0, and for all sufficiently large m, the rational map ϕm:X−−>ℙ(Wm) defined by |W_m| is birational on its image.

(II) We say that a graded linear series W• satisfies Condition (C) if

(1) for any m≫0, there exists an effective divisor F_m such that Am:=mD−Fm is ample, and

(2) for all sufficiently large t, we have

H0(X,OX(tAm))⊆Wtm⊆H0(X,OX(tmAm)).

If W• is complete, that is, Wm=H0(X,OX(mAm)) for all m≥ 0 and D is big, then it satisfies Condition (C).

Lazarsfeld and Mustată [81] proved the following.

Theorem 5.1. Let X be a smooth projective variety of dimension d. Suppose that a graded linear series W• satisfies Condition (B) or Condition (C). Then for any admissible flag Y• on X, we have

dimΔY•(W•)=dimX=d

and

Volℝn(ΔY•(W•))=1d!VolX(W•),

where

VolX(W•):=limn→∞dimWmmd/d!.

Proof. See [81, Theorem 2.13].

Remark 5.2. It is known by Lazarsfeld and Mustată ([81, Proposition 4.1]) that for a fixed admissible flag Y• on X, if D is big, then ΔY•(D) depends only on the numerical class of D. If D is not big, then it is not true (cf. [20, Remark 3.13]).

Definition 5.3. For a divisor D on X, we let

ℕ(D):=m∈ℤ+| |mD|≠∅.

For m∈ℕ(D), we let

ΦmD:X−−−>ℙdim|mD|

be the rational map defined by the linear system |mD|. We define the {\sf Iitaka dimension} of D as the following value

κ(D):=max{dimIm(ΦmD)| m∈ℕ(D)} if ℕ(D)≠∅−∞ if ℕ(D)=∅.

Definition 5.4. Let D be a divisor on X such that κ(D)≥0. A subset U of X is called a Nakayama subvariety of D if κ(D)=dimD and the natural map

H0(X,OX(mD)) →H0(U,OU(mD|U))

is injective for every non-negative integer m.

Definition 5.5. [20, Definition 3.8] Let D be a divisor on X such that κ(D)≥0. The valuative Okounkov body ΔY•val(D) associated to D is defined to be

ΔY•val(D):=ΔY•(D)⊂ℝn,  n=dimX.

For a divisor D with κ(D)=−∞, we define ΔY•val(D):=∅.

Remark 5.3. If D is big, then ΔY•val(D) coincides with ΔY•(D) for any admissible flag Y• on X.

Recently Choi, Hyun, Park and Won [20] showed the following.

Theorem 5.2. Let D be a divisor with κ(D)≥0 on a smooth projective variety X of dimension n. Fix an admissible flag Y• containing a Nakayama subvariety U of D such that Yn={x} is a general point. Then we have

dimΔY•val(D)=κ(D)

and

Volℝκ(D)(ΔY•val(D))=1κ(D)!VolX|U(D).

Proof. See [20, Theorem 3.12].

Definition 5.6. [20, Definition 3.17] Let D be a pseudo-effective divisor on a projective variety X of dimension n. The limiting Okounkov body ΔY•lim(D) of D with respect to an admissible flag Y• is defined to be

ΔY•lim(D):=limε→0+ΔY•(D+εA)⊂ℝn,

where A is any ample divisor on X. If D is not a pseudo-effective divisor, then we define ΔY•lim(D):=∅.

Definition 5.7. [20, Definition 2.11] Let D be a divisor on a projective variety X of dimension d. We defne the numerical Iitaka dimension κν(D) by

κν(D):=maxk∈ℤ+| limsupm→∞h0(X,OX(|mD|+A))mk>0

for a fixed ample Cartier divisor A if h0(X,OX(|mD|+A))≠∅ for infinitely many m>0, and we define κν(D)=−∞ otherwise.

Let D be a pseudo-effective Cartier divisor on a projective variety X of dimension n. Let V⊆X be a positive volume subvariety of D. Fix an admissible flag V• on V

V•: V=V0⊃V1⊃V2⊃⋯⊃Vn−1⊃Vn={x}.

let A be an ample Catier divisor on X. For each positive integer k, we consider the restricted graded linear series W•k:=W•(kD+A|V) of kD+A along V given by

Wm(kD+A|V)=H0(X|V,m(kD+A))  for m≥0.

We define the restricted limiting Okounkov body of a Cartier divisor D with respect to a positive volume subvariety V of D as

ΔV•lim(D):=limk→∞1kΔV•(W•k)⊆ℝκν(D).

By the continuity, we can extend this definition for any pseudo-effective ℝ-divisor.

Definition 5.8. Let D be a pseudo-effective divisor on a projective variety X of dimension n with its positive volume subvariety V⊆X. We define the restricted limiting Okounkov body ΔV•lim(D) of D with respect to an admissible flag V• to be a closed convex subset

ΔV•lim(D):=limε→0+ΔV•(D+εA)⊆ℝκν(D)↪ℝn,

where A is any ample divisor on X. If D is not a pseudo-effective divisor, then we define ΔV•lim(D):=∅.

Recently Choi, Hyun, Park and Won [20] proved the following.

Theorem 5.3. Let D be a pseudo-effective divisor on a projective variety X. Fix a positive volume subvariety V⊆X of D (see [20, Definition 2.13]). For an admissible flag V• of V, we have

dimΔV•lim(D)=κν(D)

and

Volℝκν(D)(ΔV•lim(D))=1κν(D)!VolX|V+(D).

Here VolX|V+(D) denotes the augmented restricted volume of D along V (see [20, Definition 2.2]) for the precise definition of VolX|V+(D)).

Proof. See [20, Theorem 3.20].