Nef Divisors on Moduli Spaces of Abelian Varieties 8 K. Hulek 9 9 1 Dedicated to the memory of Michael Schneider n a J 0 0 Introduction 3 Let A be the moduli space of principally polarized abelian varieties of di- 2 g v mension g. Over the complex numbers A = H /Γ where H is the Siegel g g g g 6 space of genus g and Γ = Sp(2g,Z). We denote the torodial compacti- 1 g 0 fication given by the second Voronoi decomposition by A∗g and call it the 8 Voronoi compactification. It was shown by Alexeev and Nakamura [A] that 0 A∗ coarsely represents the stack of principally polarized stable quasiabelian 7 g 9 varieties. The variety A∗ is projective [A] and it is known that the Picard g / group of A∗,g ≥ 2 is generated (modulo torsion) by two elements L and D, m g where L denotes the (Q-)line bundle given by modular forms of weight 1 o e and D is the boundary (see [Mu2], [Fa] and [Mu1] for g = 2,3 and ≥4). In g this paper we want to discuss the following - g l Theorem 0.1 Let g = 2 or 3. A divisor aL−bD on A∗ is nef if and only a g : if b ≥ 0 and a−12b ≥ 0. v i X The varieties A have finite quotient singularities. Adding a level-n g r structure one obtains spaces A (n) = H /Γ (n) where Γ (n) is the princi- a g g g g pal congruence subgroup of level n. For n ≥ 3 these spaces are smooth. However, the Voronoi compactification A∗(n) acquires singularities on the g boundary for g ≥ 5 due to bad behaviour of the second Voronoi decomposi- tion. There is a natural quotient map A∗(n) → A∗. Note that this map is g g branched of order n along the boundary. Hence Theorem (0.1) is equivalent to Theorem 0.2 Let g = 2 or 3. A divisor aL−bD on A∗(n) is nef if and g only if b ≥ 0 and a−12b ≥ 0. n This theorem easily gives the following two corollaries. 1 Corollary 0.3 If g = 2 then K is nef but not ample for A∗(4) and K is 2 ample for A∗(n), n ≥ 5; in particular A∗(n) is a minimal model for n ≥ 4 2 2 and a canonical model for n ≥ 5. This was first proved by Borisov [Bo]. Corollary 0.4 If g = 3 then K is nef but not ample for A∗(3) and K is 3 ample for A∗(n), n ≥ 4; in particular A∗(n) is a minimal model for n ≥ 3 3 3 and a canonical model for n ≥ 4. In this paper we shall give two proofs of Theorem (0.1). The first and quick onereduces theproblem viatheTorelli mapto theanalogous question for M , resp.M . SincetheTorelli mapis notsurjective forg ≥ 4this proof 2 3 cannot possibly begeneralized to higher genus. Thisis themain reason why wewanttogiveasecondproofwhichusesthetafunctions. Thisproofmakes essential use of a result of Weissauer [We]. The method has the advantage that it extends in principle to other polarizations as well as to higher g. We will also give some partial results supporting the Conjecture For any g ≥ 2 the nef cone on A∗ is given by the divisors g aL−bD where b ≥ 0 and a−12b ≥ 0. Acknowledgement It is a pleasure for me to thank RIMS and Kyoto University for their hospitality during the autumn of 1996. I am grateful to V.Alexeev andR.SalvatiManniforusefuldiscussions. ItwasSalvatiManni who drew my attention to Weissauer’s paper. I would also like to thank R. Weissauer for additional information on [We]. The author is partially supported by TMR grant ERBCHRXCT 940557. 1 Curves meeting the interior We start by recalling some results about the Kodaira dimension of A∗(n). g It was proved by Freitag, Tai and Mumford that A∗ is of general type for g g ≥ 7. The following more general result is probably well known to some specialists. Theorem 1.1 A∗(n) is of general type for the following values of g and g n ≥ n : 0 g 2 3 4 5 6 ≥ 7 . n 4 3 2 2 2 1 0 2 Proof. One can use Mumford’s method from [Mu1]. First recall that away from the singularities and the closure of the branch locus of the map H → g A (n) the canonical bundle equals g K ≡ (g+1)L−D. (1) Thisequalityholdsinparticularalsoonanopenpartoftheboundary. Ifg ≤ 4andn ≥ 3thespaces A∗(n)aresmoothandhence(1) holdseverywhere. If g g ≥ 5 then Tai [T] showed that there is a suitable toroidal compactification A˜ (n) such that all singularities are canonical quotient singularities. By g Mumford’s results from [Mu1] one can use the theta-null locus to eliminate D from formula (1) and obtains 2g−2(2g +1) 1 K ≡ (g+1)− L+ [Θ ]. (2) n22g−5 n22g−5 null (cid:18) (cid:19) We then have general type if all singularities are canonical and if the factor in front of L is positive. This gives immediately all values in the above table with the exception of (g,n) = (4,2) and (7,1). In the latter case the factor in front of L is negative. The proof that A is nevertheless of 7 general type is the main result of [Mu1]. The difficulty in the first case is that one can possibly have non-canonical singularities. One can, however, use the following argument which I have learnt from Salvati Manni: An immediate calculation shows that for every element σ ∈ Γ (2) the square g σ2 ∈ Γ (4). Hence if σ has a fixed point then σ2 = 1 since Γ (4) acts freely. g g But for elements of order 2 one can again use Tai’s extension theorem (see [T, Remark after Lemma 4.5] and [T, Remark after Lemma 5.2]). 2 Remark 1.2 The Kodaira dimension of A is still unknown. All other 6 varieties A (n) which do not appear in the above list are either rational g or unirational: Unirationality of A for g = 5 was proved by Donagi [D] g and by Mori and Mukai [MM]. For g = 4 the same result was shown by Clemens [C]. Unirationality is easy for g ≤ 3. Igusa [I2] showed that A is 2 rational. Recently Katsylo [Ka] proved rationality of M and hence also of 3 A . Thespace A (2) is rational by work of van Geemen [vG] andDolgachev 3 3 and Ortlang [DO]. A (3) is the Burkhardt quartic and hence rational. This 2 was first proved by Todd (1936) and Baker (1942). See also the thesis of Finkelnberg [Fi]. The variety A (2) has the Segre cubic as a projective 2 model [vdG1] and is hence also rational. Yamazaki [Ya] firstshowed general type for A (n), n ≥ 4. 2 3 We denote the Satake compactification of A by A . There is a natural g g map π : A∗ → A which is an isomorphism on A . The line bundle L is g g g the pullback of an ample line bundle on A which, by abuse of notation, g we again denote by L. In fact the Satake compactification is defined as the closure of the image of A under the embedding given by a suitable power g of L on A . In particular we notice that L.C ≥ 0 for every curve C on A∗ g g and that L.C > 0 if C is not contracted to a point under the map π. Let F be a modular form with respect to the full modular group Sp(2g,Z). Then the order o(F) of F is defined as the quotient of the van- ishing order of F divided by the weight of F. Theorem 1.3 (Weissauer) For every point τ ∈ H and every ε > 0 there g exists a modular form F of order o(F) ≥ 1 which does not vanish at τ. 12+ε Proof. See [We]. 2 Proposition 1.4 Let C ⊂ A∗ be a curve which is not contained in the g boundary. Then (aL−bD).C ≥ 0 if b ≥ 0 and a−12b ≥ 0. Proof. First note that L.C > 0 since π(C) is a curve in the Satake com- pactification. It is enough to prove that (aL −bD).C > 0 if a−12b > 0 and a,b ≥ 0. This is clear for b = 0 and hence we can assume that b 6= 0. We can now choose some ε > 0 with a/b > 12+ε. By Weissauer’s theorem there exists a modular form F of say weight k and vanishing order m with F(τ) 6= 0 for some point [τ] ∈ C and m/k ≥1/(12+ε). In terms of divisors this gives us that kL = mD+D , C 6⊂ D F F where D is the zero-divisor of F. Hence F k 1 L−D = D .C ≥ 0. F m m (cid:18) (cid:19) Since a/b> 12+ε ≥ k/m and L.C > 0 we can now conclude that a k L−D .C > L−D .C ≥ 0. b m (cid:18) (cid:19) (cid:16) (cid:17) 2 4 Remark 1.5 Weissauer’sresultisoptimal,sincethemodularformsoforder > 1/12haveacommonbaselocus. ToseethisconsidercurvesC inA∗ ofthe g form X(1)×{A} where X(1) is the modular curve of level 1 parametrizing elliptic curvesandAisafixedabelianvariety ofdimensiong−1. Thedegree of L on X(1) is 1/12 (recall that L is a Q-bundle) whereas it has one cusp, i.e. the degree of D on this curve is 1. Hence every modular form of order > 1/12 will vanish on C. This also shows that the condition a−12b ≥ 0 is necessary for a divisor to be nef. 2 Geometry of the boundary (I) We first have to collect some properties of the structure of the boundary of A∗(n). Recall thattheSatakecompactification isset-theoretically theunion g of A (n) and of moduli spaces A (n), k < g of lower dimension, i.e. g k A (n)= A (n)∐ ∐Ai1 (n) ∐ ∐Ai2 (n) ...∐ ∐Aig(n) . g g g−1 g−2 0 (cid:18)i1 (cid:19) (cid:18)i2 (cid:19) (cid:18)ig (cid:19) Via the map π : A∗(n)→ A (n) this also defines a stratification of A∗(n): g g g A∗(n)= A (n)∐ ∐Di1 (n) ∐ ∐Di2 (n) ...∐ ∐Dig(n) . g g g−1 g−2 0 (cid:18)i1 (cid:19) (cid:18)i2 (cid:19) (cid:18)ig (cid:19) i1 The irreducible components of the boundary D are the closures D (n) g−1 of the codimension 1 strata Di1 (n). Whenever we talk about a boundary g−1 i1 component we mean one of the divisors D (n). Then the boundary D is g−1 given by i1 D = D (n). g−1 Xi1 The fibration π : Di1 (n) → Ai1 (n) = A (n) is the universal family g−1 g−1 g−1 of abelian varieties of dimension g − 1 with a level-n structure if n ≥ 3 resp. the universal family of Kummer surfaces for n = 1 or 2 (see [Mu1]). We shall also explain this in more detail later on. To be more precise we associate to a point τ ∈ H the lattice L = (τ,1)Z2g, resp. the principally g τ,1 polarized abelian variety A = Cg/L . Given an integer n ≥ 1 we set τ,1 τ,1 L = (nτ,n1 )Z2g, resp. A = Cg/L . By K we denote the nτ,n g nτ,n nτ,n nτ,n Kummer variety A /{±1}. n,τn Lemma 2.1 Let n ≥ 3. Then for any point [τ] ∈ Ai1 (n) the fibre of π g−1 equals π−1([τ]) = A . n,τn 5 Proof. Compare [Mu1]. We shall also give an independent proof below. 2 This result remains true for n = 1 or 2, at least for points τ whose stabilizer subgroup in Γ (n) is {±1}, if we replace A by its associate g n,τn Kummer variety K . n,τn Lemma 2.2 Let n ≥ 3. Then for [τ] ∈ Ai1 (n) the restriction of Di1 (n) g−1 g−1 to the fibre π−1([τ]) is negative. More precisely 2 Dgi1−1(n)|π−1([τ]) ≡ −nH where H is the polarization on A given by the pull-back of the principal nτ,n polarization on A via the covering A → A . τ,1 n,τn τ,1 Proof. Compare [Mu1, Proposition 1.8], resp. see the discussion below. 2 Again thestatement remainstrueforn = 1or 2if wereplace theabelian variety by its Kummer variety. First proof of Theorem (0.1). We have already seen (see Remark 1.5) that for every nef divisor aL−bD the inequality a−12b ≥ 0 holds. If C is a curve in a fibre of the map A∗(n) → A (n), then L.C = 0. Lemma (2.2) g g immediately implies that b ≥ 0 for any nef divisor. It remains to show that the conditions of Theorem (0.1) are sufficient to imply nefness. For any genus the Torelli map t : M → A extends to a morphism t : M → A∗ g g g g (see [Nam]). Here M denotes the compactification of M by stable curves. g g For g = 2 and 3 the map t is surjective. It follows that for every curve C in A∗ there is a curve C′ in M which is finite over C. Hence a divisor on g g A∗, g = 2,3 is nef if and only if this holds for its pull-back to M . In the g g ∗ notation of Faber’s paper [Fa] t L = λ where λ is the Hodge bundle and ∗ t D = δ where δ is the boundary (g = 2), resp. the closure of the locus 0 0 of genus 2 curves with one node (g = 3) (cf also [vdG2]). The result follows since aλ−bδ is nef on M , g = 2,3 for a−12b ≥ 0 and b ≥ 0 (see [Fa]). 2 0 g AswehavealreadypointedouttheTorellimapisnotsurjectiveforg ≥ 4 and hence this proof cannot possibly be generalized to higher genus. The main purpose of this paper is, therefore, to give a proof of Theorem (0.1) whichdoesnotusethereductiontothecurvecase. Thiswillalsoallow usto provesomeresultsforgeneralg. Atthesametimeweobtainanindependent proof of nefness of aλ−bδ for a−12b ≥ 0 and b ≥ 0 on M for g = 2 and 0 g 3. 6 We now want to investigate the open parts Di1 (n) of the boundary g−1 i1 components D (n) and their fibration over A (n) more closely. At the g−1 g−1 same time this gives us another argument for Lemmas (2.1) and (2.2). At this stage we have to make first useof the toroidal construction. Recall that the boundary components Di1 (n) are in 1 : 1 correspondence with the g−1 maximal dimensional cusps, and these in turn are in 1 : 1 correspondence with the lines l ⊂ Qg modulo Γ (n). Since all cusps are equivalent under g the action of Γ /Γ (n) we can restrict our attention to one of these cusps, g g namely the one given by l = (0,... ,0,1). This corresponds to τ → i∞. 0 gg To simplify notation we shall denote the corresponding boundary stratum simply by D1 (n)= D (n). The stabilizer P(l ) of l in Γ is generated g−1 g−1 0 0 g by elements of the following form (cf. [HKW, Proposition I.3.87]): A 0 B 0 0 1 0 0 A B g1 = C 0 D 0, C D ∈Γg−1, (cid:18) (cid:19) 0 0 0 1     1 0 0 0 g−1 0 ±1 0 0 g2 =  0 0 1 0 , g−1  0 0 0 ±1   1 0 0 tN  g−1 M 1 N 0 g3 =  0 0 1 −tM, M,N ∈ Zg−1, g−1  0 0 0 1      1 0 0 0 g−1 0 1 0 S g4 =  0 0 1 0, S ∈ Z. g−1  0 0 0 1     We write τ = (τ ) in the form ij 1≤i,j≤g τ ··· τ τ 11 1,g−1 1g  ... ... ... = τ1 tτ2 .  τ1,g−1 ··· τg−1,g−1 τg−1,g  (cid:18) τ2 τ3 (cid:19)  τ ··· τ τ   1,g g−1,g gg    7 Then the action of P(l ) on H is given by (cf. [HKW, I.3.91]): 0 g (Aτ +B)(Cτ +D)−1 ∗ g (τ) = 1 1 , 1 τ (Cτ +D)−1 τ −τ (Cτ +D)−1Ctτ 2 1 3 2 1 2 (cid:18) (cid:19) τ ∗ g (τ) = 1 , 2 ±τ τ 2 3 (cid:18) (cid:19) τ ∗ g (τ) = 1 3 τ +Mτ +N τ′ (cid:18) 2 1 3(cid:19) where τ′ = τ +Mτ tM +Mtτ +t(M tτ )+N tM, 3 3 1 2 2 τ τ g (τ) = 1 2 . 4 τ τ +S 2 3 (cid:18) (cid:19) The parabolic subgroup P(l ) is an extension 0 1 −→ P′(l )−→ P(l ) −→ P′′(l )−→ 1 0 0 0 where P′(l ) is the rank 1 lattice generated byg . To obtain the same result 0 4 for Γ (n) we just have to intersect P(l ) with Γ (n). Note that g is in g 0 g 2 Γ (n) only for n = 1 or 2. The first step in the construction of the toroidal g compactification of A∗(n) is to divide H byP′(l )∩Γ(n) which gives amap g g 0 H −→ H ×Cg−1×C∗ g g−1 τ tτ 1 2 7−→ (τ ,τ ,e2πiτ3/n). τ τ 1 2 2 3 (cid:18) (cid:19) Partial compactification in the direction of l then consists of adding the set 0 H × Cg−1 × {0}. It now follows immediately from the above formulae g−1 for the action of P(l ) on H that the action of the quotient group P′′(l ) 0 g 0 on H × Cg−1 × C∗ extends to H × Cg−1 × {0}. Then D (n) = g−1 g−1 g−1 (H ×Cg−1)/P′′(l ) and the map to A (n) is induced by the projection g−1 0 g−1 from H ×Cg−1 to H . This also shows that D (n) → A (n) is the g−1 g−1 g−1 g−1 universal family for n ≥ 3 and that the general fibre is a Kummer variety for n = 1 and 2. Whenever n |n we have a Galois covering 1 2 π(n ,n ): A∗(n )−→ A∗(n ) 1 2 g 2 g 1 whose Galois group is Γ (n )/Γ (n ). This induces coverings D (n ) → g 1 g 2 g−1 2 D (n ), resp. D (n ) → D (n ). In order to avoid technical diffi- g−1 1 g−1 2 g−1 1 culties we assume for the moment that A∗(n) is smooth (this is the case if g 8 g ≤ 4 and n ≥ 3). In what follows we will always be able to assume that we are in this situation. Then we denote the normal bundle of D (n) in g−1 A∗(n) by N , resp. its restriction to D (n) by N . Since the g Dg−1(n) g−1 Dg−1(n) covering map π(n ,n ) is branched of order n /n along the boundary, it 1 2 2 1 follows that π∗(n ,n )n N (n ) = n N (n ). 1 2 1 Dg−1 1 2 Dg−1 2 We now define the bundle M(n) := −nN +L. Dg−1(n) This is a line bundle on the boundary component D (n). We denote the g−1 restriction of M(n) to D (n) by M(n). We find immediately that g−1 π∗(n ,n )M(n ) = M(n ). 1 2 1 2 Theadvantage ofworkingwiththebundleM(n)isthatwecanexplicitly describe sections of this bundle. For this purposeit is useful to review some basic facts about theta functions. For every element m = (m′,m′′) of R2g one can define the theta-function Θm′m′′(τ,z) = e2πi[(q+m′)τt(q+m′)/2+(q+m′)t(z+m′′)]. q∈Zg X ThetransformationbehaviourofΘm′m′′(τ,z)withrespecttoz 7→ z+uτ+u′ is described by the formulae (Θ1)–(Θ5) of [I1, pp. 49, 50]. The behaviour of Θm′m′′(τ,z) with respect to the action of Γg(1) on Hg ×Cg is given by the theta transformation formula [I1, Theorem II.5.6] resp. the corollary following this theorem [I1, p. 85]. Proposition 2.3 Let n ≡ 0mod4p2. If m′,m′′,m′,m′′ ∈ 1 Zg−1, then the 2p functions Θm′m′′(τ,z)Θm′m′′(τ,z) define sections of the line bundle M(n) on D (n). g−1 Proof. It follows from (Θ3) and (Θ1) that for k,k′ ∈ nZg−1 the following holds: Θm′,m′′(τ,z+kτ +k′)= e2πi[−12kτtk−kt(z+k′)]Θm′,m′′(τ,z). Similarly, of course, for Θm′,m′′(τ,z). Moreover the theta transformation formula together with formula (Θ2) gives Θm′,m′′(τ#,z#)= e2πi[12z(Cτ+D)−1Ctz]det(Cτ +D)1/2uΘm′,m′′(τ,z) 9 A B for every element γ = ∈ Γ (n) and C D g−1 (cid:18) (cid:19) τ# = γ(τ), z# = z(Cτ +D)−1. Here u2 is a character of Γ (1,2) with u2| ≡ 1. g−1 Γg−1(4) OntheotherhandtheboundarycomponentD (n)isdefinedbyt = 0 g−1 3 with t = e2πiτ3/n. We have already described the action of P′′(l ) on 3 0 H × Cg−1. The result then follows by comparing the transformation g−1 behaviour of (t /t2)n with respect to g and g with the above formulae 3 3 1 3 together with the fact that the line bundle L is defined by the automorphy factor det(Cτ +D). 2 This also gives an independent proof of Lemma (2.2). 3 Geometry of the boundary (II) So far we have described the stratum D (n) of the boundary component g−1 D (n) and we have seen that there is a natural map D (n) → A (n) g−1 g−1 g−1 which identifies D (n) with the universal family over A (n) if n ≥ 3. g−1 g−1 We now want to describe the closure D (n) in some detail. In order g−1 to do this we have to restrict ourselves to g = 2 and 3. First assume g = 2. ThentheprojectionD (n)→ A (n) = X0(n)extendstoaprojection 1 1 D (n) → X(n) onto the modular curve of level n and in this way D (n) is 1 1 identifiedwithShioda’smodularsurfaceS(n) → X(n). Thefibresareeither elliptic curves or n-gons of rational curves (if n ≥ 3). Similarly the fibration D (n) → A (n) extends to a fibration D (n) → A∗ whose fibres over the 2 2 2 2 boundary of A∗(n) are degenerate abelian surfaces. This was first observed 2 by Nakamura [Nak] and was described in detail by Tsushima [Ts] whose paper is essential for what follows. We shall now explain the toroidal construction which allows us to de- scribe the fibration D (n) → A∗(n) explicitly. Here we shall concentrate 2 2 on a description of this map in the most difficult situation, namely in the neighbourhood of a cusp of maximal corank. The toroidal compactification A∗(n) is given by the second Voronoi de- g composition Σ . This is a rational polyhedral decomposition of the con- g vex hull in Sym≥0(R) of the set Sym≥0(Z) of integer semi-positive (g×g)- g g matrices. For g = 2 and 3 it can be described as follows. First note that Gl(g,Z) acts on Sym≥0(R) by γ 7→ tM γM. For g = 2 we define the stan- g dard cone σ = R γ +R γ +R γ 2 ≥0 1 ≥0 2 ≥0 3 10