Compactification of the moduli of abelian varieties, Lecture at ...

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Compactification of the moduli of abelian varieties, Lecture at ...

1Compactification of the moduliof abelian varieties andMorphisms of SQ g,Kto Alexeev’s ModuliIku Nakamura(Hokkaido University)2012 November 15, Hokkaido University


2Compactification of the moduliof abelian varieties andMorphisms of SQ g,Kto Alexeev’s ModuliIku Nakamura(Hokkaido University)2012 November 15, Hokkaido University


3Table of contents• Hesse cubics• PSQAS as analogue to Tate curves• Heisenberg groups G(K), G(3)• Stability• SQ g,K : Moduli of PSQASes over Z[ζ N , 1/N ]• Another compactification SQ toricg,K• Alexeev’s moduli AP g,d• Closed immersions of SQ toricg,K• Delaunay/Voronoi decompositions∃→ SQ g,Kinto AP g,d


41 Hesse cubic curvesC(μ) :x 3 0 + x3 1 + x3 2 − 3μx 0x 1 x 2 =0(μ ∈ P 1 Z[ζ 3 ,1/3] )10.5-0.5 0-1-20202-2


5x 3 0 + x3 1 + x3 2 − 3μx 0x 1 x 2 =0if μ gets closer to ∞10.50-0.5-1-20202-2


6x 3 0 + x3 1 + x3 2 − 3μx 0x 1 x 2 =0(μ ∈ Z[ζ 3 , 1/3])if μ gets much closer to ∞10.50-0.5-1-2020-22


7x 3 0 + x3 1 + x3 2 − 3μx 0x 1 x 2 =0(μ 3 =1 or∞)It degenerates into 3 copies of P 110.50-0.5-1-2020-22


82 Moduli of cubic curvesThm 1 (classical form over C) (Hesse 1849)A 1,3 :={nonsing. cubics with 9 inflection pts}/ isom.SQ 1,3 :=A 1,3≃ C \{1,ζ 3 ,ζ3 2 }≃H/Γ(3) (H : upper half plane)= {stable cubics with 9 inflection pts}/ isom.= {Hesse cubics}/isom=id{}= A 1,3 ∪ C(μ); μ 3 =1or∞≃ P 1= {moduli of compact objects}


9We wish to extend this to aribitrary dimension1. over Z[ζ N , 1/N ] (Today) or over Z[ζ N ]2. to define a representable functor of compact obj.F := SQ g,K(fine moduli)3. to relate SQ g,K to GIT stability, (This is new)4. GIT stable objects = our model PSQASes :Projectively Stable Quasi Abelian Scheme5. to relate 3 known compactif. SQ g,K , SQ toricg,KAlexeev’s moduli A g,d


103 Moduli over Z[ζ N , 1/N ]Thm 2(a new version of the theorem of Hesse)SQ 1,3 =P 1 Z[ζ 3 ,1/3] ,the projective fine moduli(1) The universal cubic curveμ 0 (x 3 0 + x3 1 + x3 2 ) − μ 1x 0 x 1 x 2 =0where (μ 0 ,μ 1 ) ∈ SQ 1,3 =P 1 .(2) when k is alg. closed and char. k ≠3


11SQ 1,3 (k) ==A 1,3 (k) ==⎧⎫⎪⎨ closed orbit cubics⎪⎬⎪⎩ with level 3-structure /k⎪⎭ /isom.⎧⎫⎪⎨ Hesse cubics⎪⎬⎪⎩ with level 3-str. /k⎪⎭ /isom.=id.⎧⎪⎨⎫closed orbit nonsing. cubics⎪⎬⎪⎩ with level 3-str. /k⎪⎭ /isom.⎧⎫⎪⎨ nonsing. Hesse cubics⎪⎬⎪⎩ with level 3-structure /k⎪⎭ /isom.=id.


12Thm 3(N. 1999) There exists the fine moduli SQ g,Kprojective over Z[ζ N , 1/N ], N = √ |K|, Fork closed⎧⎫⎪⎨ closed orb. deg. abelian sch. /k⎪⎬SQ g,K (k) =⎪⎩ with level G(K)-structure⎪⎭ /isom.⎧⎫⎪⎨ G(K)-invariant PSQAS /k⎪⎬=⎪⎩ with level G(K)-structure⎪⎭ ,⎧⎫⎪⎨ (nonsingular) abelian schemes /k⎪⎬A g,K (k) =⎪⎩ with level G(K)-structure⎪⎭ /isom.⎧⎫⎪⎨ G(K)-inv. abelian schemes /k⎪⎬=⎪⎩ with level G(K)-structure⎪⎭


134 Comparison of three compactificationsSummary N = √ |K|, O N =Z[ζ N , 1/N ], d>0.1. SQ g,K is a proj. fine moduli over O N [N99],2. SQ toricg,Kis a proj. coarse mod. over O N [N01] [N10],3. AP g,d = {(P, G, D)} is a proper separated coarsemoduli over Z [Alexeev02],4. dim SQ g,K = dim SQ toricg,K5. dim AP g,d = g(g +1)/2+d − 1,= g(g +1)/2,6. ∃ a bij. mor. sq : SQ toricg,K→ SQ g,K[N10](SQ toricg,K )norm ≃ SQ normg,K (1)


14SQ g,K,1/N := SQ g,K : proj. over Z[ζ N , 1/N ] (1999)AP g,N : by Alexeev, over Z, dim. excessive by N − 1(2002)A g,N : by Olsson, over Z, proper separated (2008)


15Thm 4∃ a finite Galois morph. over O N , N = √ |K|,sqap : SQ toricg,K × (PN−1 \ H g,K ) → AP g,N ⊗O N(P, φ, τ ) × v ↦→ (P, Aut †0 (P ), Div(φ ∗ (v))such that for any fixed v ∈ P N−1 \ H g,K(P, φ, τ ) ↦→ (P, Aut †0 (P ), Div(φ ∗ (v))is a closed immersion of SQ toricg,K .


165 Tate curve and PSQASR:DVR, L = Frac(R) =R[1/q], q uniformizer.Tate curve G m (L)/w ↦→ qwHesse cubics at ∞ G m (L)/w ↦→ q 3 wRewrite Tate curve asG m (L)/w n ↦→ q mn w n (n ∈ Z)Hesse cubics at ∞ G m (L)/w n ↦→ q 3mn w n (n ∈ Z)The general case : B pos. def. symmetricG m (L) g /w x ↦→ q B(x,y) b(x, y)w x ,b(x, y) ∈ L × (x ∈ X, y ∈ Y )


17The usual Tate curve over CDVR RX : x 0 x 2 2 = x3 1 − x 0x 2 1 + qx3 0OrX : y 2 = x 3 − x 2 + qThe generic fibre X η : y 2 = x 3 − x 2 + q (q ≠0)The fibre X 0 : y 2 = x 2 (x − 1) for q = 0 : a limit of X qX 0 \{0, 0} =G m ,To compactify the moduli, need to find all nice limits !!


18The general case : B pos. def. symmetricThe generic fibre:G m (L) g /w x ↦→ q B(x,y) b(x, y)w x ,b(x, y) ∈ L × (x ∈ X, y ∈ Y )PSQAS is the closed fibre of it


196 Review of Theta functionsAn elliptic curve, w = e 2πiz , q = e 2πiτ/6E(τ )=C/(Z + Zτ )=C ∗ /w ↦→ wq 6 ,q = e 2πiτ/6Theta function θ k (τ,z)= ∑ m∈Z q(k+3m)2 w k+3m .The map Θ embeds E(τ )intoP 2 .Θ:E(τ ) ∋ z ↦→ [x 0 ,x 1 ,x 2 ]=[θ 0 ,θ 1 ,θ 2 ] ∈ P 2To compactify the moduliwe find the limit of the image of Θ as q → 0General case will lead us to the next definition


20Before it,recall again w = e 2πiz , q = e 2πiτ/6θ k (τ,z + 1 3 )=ζk 3 θ k(τ,z),θ k (τ,z + τ 3 )=(qw)−1 θ k+1 (τ,z),[θ 0 ,θ 1 ,θ 2 ](τ,z + τ 3 )=[θ 1,θ 2 ,θ 0 ](τ,z)σ, τ are the liftings to GL(3),z ↦→ z + 1 3 is lifted to σ(θ k)=ζ3 kθ kz ↦→ z + τ 3 is lifted to τ (θ k)=θ k+1G(3) := the group 〈σ, τ 〉The image of Θ is a Hesse cubic.


217 Heisenberg groups G(K), G(3)G(3) = 〈σ, τ 〉 acts on V , order |G(3)| = 27,V = Rx 0 + Rx 1 + Rx 2 ,σ(x i )=ζ3 i x i, τ(x i )=x i+1 (i ∈ Z/3Z)ζ 3 is a primitive cube root of 1, R ∋ ζ 3 , 1/3• x 3 0 + x3 1 + x3 2 , x 0x 1 x 2 ∈ S 3 V only are G(3)-invariant• G(3) determines x i ”uniquely” (∵ V :G(3)-irred,)• x i are classical theta over CGeneral case will lead us to the next definition


22In terms of theta, w = e 2πiz , q = e 2πiτ/6θ k (τ,z + 1 3 )=ζk 3 θ k(τ,z),θ k (τ,z + τ 3 )=(qw)−1 θ k+1 (τ,z),[θ 0 ,θ 1 ,θ 2 ](τ,z + τ 3 )=[θ 1,θ 2 ,θ 0 ](τ,z)σ, τ are the liftings to GL(3),z ↦→ z + 1 3 is lifted to σ(θ k)=ζ3 kθ kz ↦→ z + τ 3 is lifted to τ (θ k)=θ k+1G(3) := the group 〈σ, τ 〉


238 Definition of PSQASR : DVR, q a uniformizer of R,k(0) = R/m, k(η) =R[1/q] : the fraction field of RSuppose (G η ,L η ) : abelian variety over k(η)(G, L) is the (connected) Néron model of (G η ,L η )Let λ(L η ):G η → t G η = Pic 0 (G η )( t G η , t L η ) dual AV, t G η = Pic 0 (G η ).( t G, t L) : the (connected) Néron model of ( t G η , t L η )Suppose G 0 a split torus over k(0),Then ( t G 0 , t L 0 ) is a split torus over k(0)


24For the Tate curve over CDVR RThe generic fibre G η : y 2 = x 3 − x 2 + q (q ≠0)The fibre X 0 : y 2 = x 2 (x − 1) for q = 0 : a limit of X qX 0 \{0, 0} =G m ,This is the key assumption G 0 a split torus


25x 3 0 + x3 1 + x3 2 − 3μx 0x 1 x 2 =0(μ 3 =1 or∞)It degenerates into 3 copies of P 1μ = ∞, x 0 x 1 x 2 = 0 contains G m × Z/3ZThis is the key assumption G 0 a split torus10.50-0.5-1-2020-22


26Definition of PSQASR : DVR, q a uniformizer of R,k(0) = R/m, k(η) =R[1/q] : the fraction field of RSuppose (G η ,L η ) : abelian variety over k(η)(G, L) is the (connected) Néron model of (G η ,L η )Let λ(L η ):G η → t G η = Pic 0 (G η )( t G η , t L η ) dual AV, t G η = Pic 0 (G η ).( t G, t L) : the (connected) Néron model of ( t G η , t L η )Suppose G 0 a split torus over k(0),Then ( t G 0 , t L 0 ) is a split torus over k(0)


27Let X = Hom(G 0 , G m ), Y = Hom( t G 0 , G m ).Hence X ≃ Z g , Y ≃ Z g ,λ(L η ) extends, ∃ a surjection G 0 → t G 0Hence Y : a sublattice of X, [X : Y ] < ∞.K η := ker λ(L η ), N := |K η |.K:=the closure of K η . May assume Over Z[ζ N , 1/N ]K ≃ (X/Y ) ⊕ (X/Y ) ∨ ,This finite group helps us to take up the necessary data


28From G and K we can construct• G(K) : Heisenberg group scheme1 → μ N → G(K) → K → 0 (exact)(a, z, α) · (b, w, β) =(abβ(z),z+ w, α + β),• R[X/Y ]=⊕ x∈X/Y Rv(x) (group alg. of X/Y )v(0) = 1, v(x + y) =v(x)v(y)• G(K) acts on R[X/Y ]by(a, z, α) · v(x) =aα(x)v(z + x)a, b ∈ μ N ; z, x ∈ (X/Y ); α, β ∈ (X/Y ) ∨


29Facts. G : conn. Néron model of G η ,K η := ker(λ(L η )) ≃ (X/Y ) ⊕ (X/Y ) ∨ ,• V := H 0 (G, L): finite R-free, G(K)-irreducible• V = H 0 (G, L) ≃ R[X/Y ] as G(K)-module• H 0 G(K)-isom(G, L) ∋∃θ x ←→ v(x) ∈ R[X/Y ] gp algθ x can be thought as ”classical theta”Idea: Find the limit of the image [θ x ] x∈X/Y


30Let G for : the formal completion of G along G 0Key Fact:G for ≃ (G g m,R ) forFourier expansion of θ x (x ∈ X/Y )onG for :θ x = ∑ y∈Ya(x + y)wx+ya(x + y) : Fourier coeff. of θ xcalled Faltings-Chai’s degeneration data of (G, L)• B(x, y) :=val q (a(x + y)a(x) −1 a(y) −1 )ispos. def.


31generalized Tate curvesThe general case : B pos. def. symmetricThe generic fibre:G m (L) g /w x ↦→ q B(x,y) b 0 (x, y)w x ,b 0 (x, y) ∈ L × (x ∈ X, y ∈ Y )PSQAS is the closed fibre of a gener. Tate curve


32We construct a canonical gen. of Tate curves.˜R := R[a(x)w x ϑ, x ∈ X],ϑ:deg oneProj( ˜R) : locally of finite type over RX : the formal completion of Proj( ˜R)The Quotient X /Y is a degenerating family of AV(X /Y, O X /Y (1)) is a generalization of Tate curves


33Grothendieck (EGA) guarantees∃ a projective R-scheme (Z, O Z (1))s.t. the formal completion Z for of ZZ for ≃X/Y , (Z η ,O Zη (1)) ≃ (G η ,L η )(the stable reduction theorem)The central fiber (Z 0 ,O Z0 (1)) is our (P)SQAS.Projectively Stable Quasi Abelian SchemeG(K) acts on (Z, O Z (1))


34Summary Let R be CDVR over Z[ζ N , 1/N ]• There is a natural choice of θ x ∈ H 0 (G, L)• a(x + y), y ∈ Y is Fourier coeff of θ x ,x ∈ X/Y• all a(x) recover the given G η over k(η) := Frac(R)• There is an extention X /Y of G η to R so that(a) it is a canonical generalization of Tate curves,(b) G(K) acts on (X /Y, O X /Y (1))(c) hence G(K) acts on (Z, O Z (1))(d) the closed fibre (Z 0 ,O Z0 (1)) is a PSQAS.


35Exam 1 g =1,X =Z,Y = 3Z.X = Proj( ˜R), a(x) =q x2 ,(x ∈ X)S −3✲S −3✲V −2 V −1 V 0 V 1 V 2 V 3 V 4✇ ✇ ✇ ✇ ✇ ✇ ✇❚❚✔ ✔✔✔✔2❚❚❚❚❚❚❚❚2✔ ✔✔✔✔✔✔✔ ❚❚❚2X 0 /Y


36RecallThm 5 Over Z[ζ 3 , 1/3]A 1,3 :={nonsing. cubics with 9 inflection pts}/ isom.SQ 1,3 :=A 1,3= {stable cubics with 9 inflection pts}/ isom.= {Hesse cubics}/isom=id{}= A 1,3 ∪ C(μ); μ 3 =1or∞≃ P 1 .Hesse cubics are PSQASes in dimension one, level 3.


37We wish to extend this to arbitrary dimension1. over Z[ζ N , 1/N ]orover Z[ζ N ]2. to define a representable functor of compact obj.F := SQ g,K(fine moduli)3. to relate to GIT stability, that is,to aim at F (k) =GIT stable objects for k alg. closed


38SQ g,K,1/N := SQ g,K : proj. over Z[ζ N , 1/N ] (1999)AP g,N : over Z, dim. excessive by N − 1 (2002)Olsson : over Z, nonseparated nonproper stack (2008)Olsson uses the same model as ours (Alexeev-Nakamura’smodel)We prefer to separated moduli.It is easy to construct nonseparated stack moduli.


399 Separatedness of the moduliThere are difficulties never seen in dimension one• Classical level structure = base of n-divison points,• Singular limits of Abelian varieties are very reducible• Classical level str. gives non-separated moduli• We need to prove in any dimension,Lemma. (Valuative Lemma for Separatedness)R : DVR, L = Frac(R), X, Y ∈ F (R).If X L ≃ Y L , then X ≃ Y . In other words,Isom. over L implies isom. over R.


40• separated = Hausdorff,(e.g. if X projective, thenseparated)• X: non-separated = non Hausdorff,• If non-Hausdorff, then ∃ P n ∈ X (n =1, 2, ···),P = lim P n , Q = lim P n . But P ≠ Q• This really happens in geometry.


41Example R : DVR, q : uniformizer of R, L = R[1/q],E, E ′ : elliptic curves over RE : y 2 = x 3 − q 6 , E ′ : Y 2 = X 3 − 1Let us consider P n := E L ,Q n := E ′ LP n = Q n , i.e. E L ≃ EL′becauseE L :(y/q 3 ) 2 =(x/q) 3 − 1,E L ′ : Y 2 = X 3 − 1


42Example R : DVR, q : uniformizer of R, L = R[1/q],E, E ′ : elliptic curves over RE : y 2 = x 3 − q 6 , E ′ : Y 2 = X 3 − 1Let us consider P n := E L ,Q n := E ′ LP := E 0 = lim E L , Q := E ′ 0 = lim E′ LP n = Q n , i.e. E L ≃ E ′ LButP ≠ QP := E 0 : y 2 = x 3 , Q := E ′ 0 : Y 2 = X 3 − 1


43To overcome the difficulty of level str/n-div. pts :• Non-abelian Heisenberg gp. G := G(K)• New level str. = Framing of irred. reps. of G• To prove Val. Lemma for Separatedness, we useSchur’s Lemma over R. Let |G| = N,R : a ring over Z[ζ N , 1/N ], V : free R-mod.V : irr. G-mod. of wt one, (⇒ G ⊂ GL(V ⊗R))Let h ∈ GL(V ⊗ R). If gh = hg for ∀ g ∈ G,then h is scalar.


44Summary• Separatedness of the modulifollows from G(K)-Irreducibility of V = H 0 (X, L),(X, L) =(Z 0 ,O Z0 (1)) : any PSQAS, level N ≥ 3if K ≃ ker(λ(L) :G η → G t η (dual)).


45We re-start withThm 6 Over Z[ζ 3 , 1/3]A 1,3 :={nonsing. cubics with 9 inflection pts}/ isom.A 1,3 :={stable cubics with 9 inflection pts}/ isom.= {Hesse cubics}/isom=id{}= A 1,3 ∪ C(μ); μ 3 =1or∞≃ P 1 .We convert it into G(3)-equivariant theoryG(3): Heisenberg group of level 3


4610 Heisenberg groups G(K), G(3)G(3) = 〈σ, τ 〉 acts on V , order |G(3)| = 27,V = Rx 0 + Rx 1 + Rx 2 ,σ(x i )=ζ3 i x i, τ(x i )=x i+1 (i ∈ Z/3Z)ζ 3 is a primitive cube root of 1, R ∋ ζ 3 , 1/3Fact• x 3 0 + x3 1 + x3 2 , x 0x 1 x 2 ∈ S 3 V only are G(3)-invariant• G(3) determines x i ”uniquely” (∵ V :G(3)-irred,)• x i are classical theta over C


47Summary G(K) : Heisenberg gp. e.g. G(3)• G(K) chooses a basis of V = H 0 (X, L), X:PSQAS• G(K) chooses a basis of H 0 (G, L), G:Néron model• G(K) determines Faltings-Chai degeneration data• G(K) extends G η to define (Z, O Z (1)), Z = X /Y• Separatedness of the modulifollows from G(K)-Irreducibility of V = H 0 (X, L),X : any PSQAS, level N ≥ 3


4811 The space of closed orbitsXGthe set of geometric objectsthe group of isomorphismsx, x ′ are isom. G-orbits are the same O(x) =O(x ′ )X psX ssX ss //Gthe set of properly-stable objectsthe set of semistable objects”compact moduli


49Exam 2 Action on C 2 of G =G m (= C ∗ ),C 2 ∋ (x, y) ↦→ (αx, α −1 y) (α ∈ G m )What is the quotient of C 2 by G ?• Simple answerthe set of G-orbits (×)• AnswerSpec(the ring of all G-invariant poly.)()• t := xy is the unique G-inv. !C 2 //G := Spec C[t] ={t ∈ C}But this is different from ”the set of G-orbits”.• C 2 //G = {t ∈ C} is the set of all closed orbits.


50xy =0✻O(d, 1)11O(c, 1)∼ = {t ∈ C}5✲xy =0(c >0,d


51Def 7 The same notation as before. Let p ∈ X.(1) semistable if ∃ G-inv. homog. poly. F , F (p) ≠0,(2) Kempf-stable (= closed orbit)if the orbit O(p) is closed in X ss ,(3) properly-stable if (2) and Stab(p) finite.Remstable =⇒ closed orbit =⇒ semistable


52Thm 8(Seshadri,Mumford) G : reductive, acting ona scheme X, (e.g. G = G m ). Let X ss = the set ofsemistable points. Then• X ss //G := Spec(all G-inv.) = the set of closed orbits.• X ss //G is a scheme, X ps //G is also a scheme,• X ss //G compactifies X ps //G.RemThe set of points with closed orbits is not analgebraic subscheme.


53Thus we consider only those objects with closed orbitsAs its consequence we will see• Abelian varieties have closed orbits (Kempf), and• our PSQASes have closed orbits,Conversely• Any degenerate abelian scheme with closed orbitis one of our PSQASes• There is a simple characterization of our PSQASes,• This characterization enables us to compactifythe moduli of abelian varieties.


5412 Stable curves of Deligne-MumfordDef 9C is a stable curve of a genus g if(1) connected projective reduced with finite autom.,(2) the singularities of C are like xy =0(3) dim H 1 (O C )=gLet M g : moduli of stable curves of genus g,M g : moduli of nonsing. curves of genus g.Thm 10M g compactifies M g(Deligne-Mumford 1969)


Definition of stable curves is irrelevant to GIT stabilityNevertheless55Thm 11The following are equivalent(1) C is a stable curve (moduli-stable)(2) any Hilbert point of Φ |mK| (C) isGIT-stable(3) any Chow point of Φ |mK| (C) isGIT-stable(1)⇔(2) Gieseker 1982 (before Mumford 1977)(1)⇔(3) Mumford 1977 (suggested by Gieseker 1982)


5613 Stability of cubic curvesCUBIC CURVESSTABILITY STAB GP.smooth elliptic stable finite3-gon closed orbit 2-dima line+a conic (transv.) semistable 1-dimirred. with a node semistable finiteothers unstable 1-dim


57Thm 12For a cubic C, the following cond. are equiv.(1) C has a closed SL(3)-orbit in (S 3 V ) ss (2) C is a Hesse cubic curve, that is, G(3)-invariant(3) C is either smooth elliptic or a 3-gon


5814 Stability in higher-dim.Thm 13 (Kempf) (A, L) an abelian variety,V = H 0 (A, L) very ample, w:=Hilbert point of (A, L).Then SL(V )w is closed in P ss : the semistable locus ofa big proj. space.Thm 14(N.1999)(X, L) : PSQAS of level G(K),V = H 0 (X, L) very ample. Thenany Hilbert point of (X, L) has a closed SL(V )-orbit.


59Thm 15(N.1999)Assume (X, L) is a limit of abelian varieties Awith ker(λ(L)) = K, λ(L) :A → A t (dual)Then the following are equivalent:(1) X has a closed SL(V )-orbit (GIT-stable)(2) X is invariant under G(K) (G(K)-stable)(3) X is one of our PSQASes (moduli-stable)


60To be more precise,Thm 16(N.1999)Assume (X, L) is a limit of AV A’s with ker(λ(L)) = KThen the following are equivalent:(1) The m-th Hilbert point of X has a closed SL(V )-orbit in P( M ∧ S m V ) ss(GIT-stable)(2) X is invariant under G(K) (G(K)-stable)(3) X is one of our PSQASes (moduli-stable)where M := dim H 0 (X, mL).


61Thm 17For cubics the following are equiv:(1) it has a closed SL(3)-orbit (GIT-stable)(2) it is a Hesse cubic, that isG(3)-inv. (G(3)-stable)(3) it is smooth ell. or a 3-gon. (moduli-stable)Thm 18Let X be a degenerate AV. The following areequiv. under natural assump.:(1) it has a closed SL(V )-orbit (GIT-stable)(2) X is G(K)-inv (G(K)-stable)(3) it is a PSQAS (p.20) (moduli-stable)


62Thus we see• Abelian varieties have closed orbits (Kempf), and• our PSQASes have closed orbits,Conversely• Any degenerate abelian scheme with closed orbitis one of our PSQASes• X is our PSQAS iff X is G(K)-stable,• This characterization will compactifythe moduli of abelian varieties.


63The characterization of PSQASes will compactifythe moduli of abelian varieties. We recall”Closed orbit” is not a Zariski open/closed condition.Exam 3Let G := {(s, t, u) ∈ (G m ) 3 ; stu =1}C a,b,c : ax 3 0 + bx3 1 + cx3 2 − x 0x 1 x 2 =0.G acts on A 3 : (a, b, c) ↦→ (sa, tb, uc)A 3Closed (G m ) 2 -orbit iff abc ≠ 0 or (a, b, c) =(0, 0, 0).


6415 Moduli over Z[ζ N , 1/N ]Thm 19(a new version of the theorem of Hesse)SQ 1,3 =P 1 Z[ζ 3 ,1/3] ,the projective fine moduli(1) The universal cubic curveμ 0 (x 3 0 + x3 1 + x3 2 ) − μ 1x 0 x 1 x 2 =0where (μ 0 ,μ 1 ) ∈ SQ 1,3 =P 1 .(2) when k is alg. closed and char. k ≠3


65SQ 1,3 (k) ==A 1,3 (k) ==⎧⎫⎪⎨ closed orbit cubics⎪⎬⎪⎩ with level 3-structure /k⎪⎭ /isom.⎧⎫⎪⎨ Hesse cubics⎪⎬⎪⎩ with level 3-str. /k⎪⎭ /isom.=id.⎧⎪⎨⎫closed orbit nonsing. cubics⎪⎬⎪⎩ with level 3-str. /k⎪⎭ /isom.⎧⎫⎪⎨ nonsing. Hesse cubics⎪⎬⎪⎩ with level 3-structure /k⎪⎭ /isom.=id.


66Thm 20(N. 1999) There exists the fine moduli SQ g,Kprojective over Z[ζ N , 1/N ], N = √ |K|, Fork closed⎧⎫⎪⎨ closed orb. deg. abelian sch. /k⎪⎬SQ g,K (k) =⎪⎩ with level G(K)-structure⎪⎭ /isom.⎧⎫⎪⎨ G(K)-invariant PSQAS /k⎪⎬=⎪⎩ with level G(K)-structure⎪⎭ ,⎧⎫⎪⎨ (nonsingular) abelian schemes /k⎪⎬A g,K (k) =⎪⎩ with level G(K)-structure⎪⎭ /isom.⎧⎫⎪⎨ G(K)-inv. abelian schemes /k⎪⎬=⎪⎩ with level G(K)-structure⎪⎭


67Summary G(K) : Heisenberg gp. e.g. G(3)(A)H 0 (X, L) isG(K)-irred for X: PSQAS• (A) implies Stability of X with L very ample,• (A) implies Separatedness of the moduli,• (A) gives a simple characterization of PSQASes,• G(K) finds a compact separated moduli SQ g,K


6816 The Second Compactification over Z[ζ N , 1/N ]Recall Grothendieck (EGA) guarantees∃ a projective R-scheme (Z, O Z (1))s.t. the formal completion Z for of ZZ for ≃X/Y , (Z η ,O Zη (1)) ≃ (G η ,L η )The central fiber (Z 0 ,O Z0 (1)) is our (P)SQAS.The normalization Z norm of Z with Z norm0reducedgives a bit different central fiber(Z norm0,O Znorm(1)), we call it TSQAS.0


69Thm 21 (N. 2010) over Z[ζ N , 1/N ],∃ another cano. compactif. SQ toricg,K:coarse moduli of TSQASes with level-G(K) str.∃ cano. bij. birat. morphismsq : SQ toricg,K→ SQ g,K(P, φ, τ ) ↦→ (Q, φ Q ,τ Q ),Q := Proj(Sym(φ))when any generic fibre of P is an abelian var.CorollaryThe normalizations of SQ toricg,Kand SQ g,K are isom.


70Recall (P, φ, τ ) ∈ SQ toricg,K• P :TSQAS=modified PSQAS,• φ : P → P N−1 =P(k[H ∨ ]) is a finite morphism• L = φ ∗ (O P N−1(1)),• H 0 (P, L) φ∗≃ k[H ∨ ]=H 0 (O P N−1(1))• τ : a compatible action of G(K) on the pair (P, L)• τ on P = translation by K when P = A : AV


71(Q, φ Q ,τ Q ) ∈ SQ g,K• Q:PSQAS,• φ Q : Q → P N−1 =P(k[H ∨ ]) is a closed immersion• L Q = φ ∗ (O P N−1(1)),• H 0 (Q, L Q ) ≃ H 0 (P, L) φ∗≃ k[H ∨ ]=H 0 (O P N−1(1))• τ Q : a compatible action of G(K) on the pair (Q, L Q )• τ Q on Q = translation by K when Q = A : AV


72Definition of sq : For (P, L, φ, τ ) ∈ SQ toricg,K (T )Suppose (P, L, φ, τ )isaT -TSQASsuch that any generic fibre is AV.Then let Q = φ(P ) := Proj(Sym(φ))Can define (Q, L Q ,φ Q ,τ Q ) T -PSQAS, Thenthe morphism sq issq(P, L, φ, τ )=(Q, L Q ,φ Q ,τ Q ) ∈ SQ g,K (T )


7317 Comparison of three compactificationsSummary N = √ |K|, O N =Z[ζ N , 1/N ], d>0.1. SQ g,K is a proj. fine moduli over O N [N99],2. SQ toricg,Kis a proj. coarse mod. over O N [N01] [N10],3. AP g,d = {(P, G, D)} is a proper separated coarsemoduli over Z [Alexeev02],4. dim SQ g,K = dim SQ toricg,K5. dim AP g,d = g(g +1)/2+d − 1,= g(g +1)/2,6. ∃ a canonical bij. birat. morphism [N10]sq : SQ toricg,K→ SQ g,K


74Alexeev’s moduli AP g,d = {(P, G, D)}• P is semi-normal proj. with L ample line bundle• G semi-abelian acting on P with extra cond.• D ∈ H 0 (P, L) a Cartier divisor• D contains no G-orbits• dim AP g,d = dim A g + d − 1.


75k alg. closedSQ 1,K , K =(Z/3Z) 2 , RoughlySQ 1,K (k) ={C a nonsing. cubic or a 3-gon cubic}AP 1,3 (k) ={(C, G, D)}C nonsingular elliptic or a 3-gon,or a conic plus a line, rational with a nodeG = C (elliptic) or G m , D ∈ H 0 (C, L), degree D =3.


76To define a morphism from SQ 1,K to AP 1,3is equivalent to the followingFor a givena flat family over T(C, φ, τ ) ∈ SQ 1,K (T )always ! construct (G, D) so that(C, G, D) ∈ AP 1,3 (T )


77Problem: Construct G and Find DFor almost all v ∈ k[Z/3Z],(P, φ, τ ) × v↦→ (P, Aut †0 (P ), Div(φ ∗ (v))Need to proveAny T -TSQAS has a flat group scheme actionThis is done in generalThm 22If (P, L) isanS-flat TSQAS, thenAut †0S(P )isS-flat semi-abelian group scheme


78Thm 23∃ a finite Galois morph. over O N , N = √ |K|,sqap : SQ toricg,K × (PN−1 \ H g,K ) → AP g,N ⊗O N(P, φ, τ )× ↦→ (P, Aut †0 (P ), Div(φ ∗ (v))such that for any fixed v ∈ P N−1 \ H g,K(P, φ, τ ) ↦→ (P, Aut †0 (P ), Div(φ ∗ (v))extending an injec-is an injective morphism of SQ toricg,Ktive immersion of A toricg,K .• P N−1 =P(O N [H ∨ ] ∨ ), v ∈O N [H ∨ ].• H g,K is a hypersurf. of P N−1 of deg. known.• dim SQ toricg,K + N − 1 = dim AP g,N.


79SQ g,K,1/N := SQ g,K : over Z[ζ N , 1/N ]AP g,N : Alexeev, over Z, no level str.A g,N : Olsson, over Z, no level str.


8018 The shape of PSQASes — Delaunay decompositions”Limits of theta functions are described by theDelaunay decomposition.”PSQAS is a geometrization of limit of thetasPSQAS is a generalization of 3-gons.which is described by the Delaunay decomposition.


81PSQAS : a generalization of Tate curve, R:DVRTate curve G m (R)/w ↦→ qwHesse cubics at ∞ G m (R)/w ↦→ q 3 wRewrite Tate curve as G m (R)/w n ↦→ q mn w n (m ∈ Z)Hesse cubics at ∞ G m (R)/w n ↦→ q 3mn w n (m ∈ Z)The general case : B pos. def. symmetricG m (R) g /w x ↦→ q B(x,y) b(x, y)w x ,b(x, y) ∈ R × (x ∈ X, y ∈ Y )


82Let X =Z g , B a positive symmetric on X × X.‖x‖ = √ B(x, x) : a distance of X ⊗ R (fixed)Def 24Let α ∈ X R . a Delaunay cell D(α) : the convexclosure of points of X closest to α.Exam 41-dim. B(x, y) =2xy, X/Y =Z/nZ,then PSQAS Z 0 is an n-gon of P 12 2 2 2 2 2 2


• All Delaunay cells for a B form a Delaunay decomp.• Each PSQAS (its scheme struture) and its decompositioninto torus orbits (its stratification)are described by Delaunay decomp.• Each pos. symm. B defines a Delaunay decomp.• Different B can yield the same Delaunay decomp.and the same PSQAS.83


⎛ ⎞⎜Exam 5 B = ⎝ 10 ⎟⎠01Z 0 := X 0 /Y is a union of P 1 × P 1844 4 4 4 4 44 4 4 4 4 44 4 4 4 4 44 4 4 4 4 4


Exam 6 B =⎛ ⎞⎜⎝ 2 −1 ⎟⎠−1 2857 7 77✁ ✁✁✁✁7✁ ✁✁✁✁✟✟❅ ✟✟❅✟✟ ❅❅ ✟✟ ❅❅7✟7✟ ✟✟✟7 7 7✟✁7✁✁✁✁✟7✁ ✁✁✁✁✟ ❅✟✟✟✟ ❅❅❅7✟✟ ❅❅7✟ ✟✟✟7 ✁7 ✁✁✁7✁✁✁✁✁✁


861 1 1 1 11 1 1 1 11 1 1 1 1 1 1 1 1 1 1. This (mod Y ) is a PSQAS.It is a union of P 2 , each triangle stands for P 2 ,2. each line segment is a P 1 ,twoP 2 intersect along P 13. six P 2 meet at a point,locally k[x 1 , ··· ,x 6 ]/(x i x j , |i − j| ≥2)


871 ❅ 1✟ ✟ 1 ❅ 1✟ ✟ 1 ❅1✟ ✟ 1 ❅ 1✟ ✟11 1 1 1 ✟✟ ❅ ✁✁✁ ✁ ✁✁✁ ✁✁✁ ✁✁1 ✟✟❅1 ✟✟❅1 ✟✟❅1✟1✟1✟1 1✟ ✟11 1 1 1✟✟1 ❅ ✁✁✁ ✟✁ ✁✁✟✁ ✁✁✟✁ ✁✁1✟✟1 ❅ 1✟✟1 ❅ 1✟✟1 ❅ 1✟ ✟11 1 1 1✟✟ ❅ ✁✁✁ ✁ ✁✁✁ ✁✁✁ ✁✁✟✟❅ ✟✟❅ ✟✟❅ 1 1✟1 1✟1 1✟1 1✟ ✟11111 ✟✟ ✁✁✁✁✁ ✁ ✁✁ ✁ ✁✁ ✁ ✁✁ ✟ ❅✟✟✟ ❅✟✟✟ ❅✟✟ ❅Red one is the decomp. dual to the Delaunay decomp.called Voronoi decomp.


88❅ ✟1 ✟ ❅ ✟1 ✟ ❅ 1✟ ✟ ❅ 1✟ ✟1 1 1 1✟✟ ❅ ✁✁✁ ✁ ✁✁✁ ✁✁✁ ✁✁✟✟❅ ✟✟❅ ✟✟❅ ✟1 ✟1 1✟1✟ ✟✁1 1 1 1✟✟ ❅ ✁✁✁ ✟✁ ✁✁✟✁ ✁✁✟✁ ✁ 1✟✟ ❅ 1✟✟ ❅ 1✟✟ ❅ 1✟ ✟1 1 1 1✟✟ ❅ ✁✁✁ ✁ ✁✁✁ ✁✁✁ ✁✁✟✟❅ ✟✟❅ ✟✟❅ 1✟1✟1✟1✟ ✟1 1 1 1✟✟ ✁✁✁✁✁ ✁ ✁✁ ✁ ✁✁ ✁ ✁✁ ✟❅ ✟✟✟ ❅ ✟✟✟ ❅ ✟✟ ❅


89❅ ✟1 ✟ ❅ ✟1 ✟ ❅ 1✟ ✟ ❅ 1✟ ✟1 1 1 1✟✟ ❅ ✁✁✁ ✁ ✁✁✁ ✁✁✁ ✁✁✟✟❅ ✟✟❅ ✟✟❅ ✟1 ✟1 1✟1✟ ✟✁1 1 1 1✟✟ ❅ ✁✁✁ ✟✁ ✁✁✟✁ ✁✁✟✁ ✁ 1✟✟ ❅ 1✟✟ ❅ 1✟✟ ❅ 1✟ ✟1 1 1 1✟✟ ❅ ✁✁✁ ✁ ✁✁✁ ✁✁✁ ✁✁✟✟❅ ✟✟❅ ✟✟❅ 1✟1✟1✟1✟ ✟1 1 1 1✟✟ ✁✁✁✁✁ ✁ ✁✁ ✁ ✁✁ ✁ ✁✁ ✟❅ ✟✟✟ ❅ ✟✟✟ ❅ ✟✟ ❅Voronoi decomposition


90Def 25D : for Delaunay cellsV (D) :={λ ∈ X ⊗ Z R; D = D(λ)}We call it a Voronoi cellV (0) = {λ ∈ X ⊗ Z R; ‖λ‖ ≦ ‖λ − q‖, (∀q ∈ X)} ✁4 ✁ 4 4✟4✟ ✟ ❅ ✁ ✁✁ ✟✁ ✁✁✟✟ ❅4 ❅4✟4 4✟ ❅4✟4 4✟ ✟✁ ✁✁✁✟✁ ✁✁✁❅✟✟ ❅❅4 4✟❅✁4 ✟ ✟✟✁4✁✁✁ ✁ ✁ ✁ This is a crystal of mica.


For B =⎛ ⎞100⎜010⎟⎝ ⎠00191We get V (0), a cube (salt),For B =⎛ ⎞1 0 0⎜0 2 −1⎟⎝ ⎠0 −1 2then we get a hexagonal pillar (calcite)and then


B =⎛⎞2 −1 0⎜−1 2 −1⎟⎝⎠0 −1 292A Dodecahedron (Garnet)


B =⎛⎞2 −1 0⎜−1 3 −1⎟⎝⎠0 −1 293Apophyllite KCa 4 (Si 4 O 10 ) 2 F · 8H 2 O


B =⎛⎞3 −1 −1⎜−1 3 −1⎟⎝⎠−1 −1 394A Trunc. Octahed. — Zinc Blende ZnS

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