Compactification of the moduli space of abelian varieties, Kyoto ...

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6 IKU NAKAMURA4.1. Limit objects. First we note• Any PSQAS is a scheme-theoretic limit of the images of AV by thetafunctions. It is also a compactification of a generalized Tate curve.Let R be a CDVR, and k(η) the fraction field of R. We start with anabelian scheme (G η , L η ) and a polarization morphism λ(L η ):G η → G t η.Let K η = ker(L η ) the finite group scheme, and G(K η ) := Aut(L η /G η ): theautom. gp of the pair (G η , L η ) linear in the fibers of L η over G η .For simplicity, we assume the characteristic of k(0) = R/m R is prime torank K η . Then there exists a finite symplectic abelian group K such thatK η ≃ K and G(K η ) ≃G(K) by some base change1 → G m →G(K) → K → 0 (exact)Theorem 4.2. (A refined version of Alexeev-Nakamura’s stable reductiontheorem) ([AN99], [N99]) For an abelian scheme (G η , L η ) and a polarizationmorphism λ(L η ) : G η → G t η over k(η), there exist proper flat projectiveschemes (Q, L Q ) (PSQAS) and (P,L P ) (TSQAS) over R, by a finite basechange if necessary, such that(o) (Q η , L η ) ≃ (P η , L η ) ≃ (G η , L η ),(i) (P,L P ) is the normalization of (Q, L Q ),(ii) P 0 is reduced,(iii) if e min (K) ≥ 3, then L Q is very ample, and in general, (Q, L Q ) is anétale quotient of some PSQAS (Q ∗ , L Q ∗) with L Q ∗ very ample,(iv) G(K) acts on (Q, L Q ) and (P,L P ) extending the action of it on (G η , L η ).• The above theorem proves that the moduli is proper,• (Q 0 , L 0 ): PSQAS — projectively stable quasi-abelian scheme,• (P 0 , L 0 ): TSQAS — torically stable quasi-abelian scheme (= variety),• In dim. one, any PSQAS=TSQAS is a smooth elliptic or an N-gon,• The next theorem proves that the moduli is separated.Theorem 4.3. ([N99],[N10],[N13]) Suppose e min (K) ≥ 3. Then (Q, L) and(P,L) are uniquely determined by (G η , L η ).5. PSQAS and TSQAS in low dimension5.1. Hesse cubics and thetas. Now we calculate the limit of [θ 0 ,θ 1 ,θ 2 ]as q → 0.Let R be a CDVR and I = qR. Then the power series θ k convergeI-adically: First we shall show a rather strange computation, which may
COMPACTIFICATION OF THE MODULI SPACE OF ABELIAN VARIETIES 7embarass you.Hence in P 2θ 0 (q, w) = ∑ q 9m2 w 3mm∈Z=1+q 9 w 3 + q 9 w −3 + q 36 w 6 + ··· ,θ 1 (q, w) = ∑ q (3m+1)2 w 3m+1m∈Z= qw + q 4 w −2 + q 16 w 4 + ··· ,θ 2 (q, w) = ∑ q (3m+2)2 w 3m+2m∈Z= qw −1 + q 4 w 2 + q 16 w −4 + q 25 w 5 + ··· .lim [θ 0 ,θ 1 ,θ 2 ](q, w)] = [1, 0, 0]q→0The elliptic curves converge to one point ? This looks strange.To understand this, we need to understand something much deeper, Néronmodel. We cannot explain it in detail. Instead we show a necessary modificationof the above computation. The reason why we got the above is thatwe treated w as constant.Let w = q −1 u for u ∈ R \ I and u = u mod I. Then the power series θ kconverge I-adically:Hence in P 2Similarlyθ 0 (q, q −1 u)= ∑ q 9m2−3m u 3mm∈Z=1+q 6 u 3 + q 12 u −3 + q 30 u 6 + ··· ,θ 1 (q, q −1 u)= ∑ q (3m+1)2−3m−1 u 3m+1m∈Z= u + q 6 u −2 + q 12 u 4 + ··· ,θ 2 (q, q −1 u)= ∑ q (3m+2)2−3m−2 u 3m+2m∈Z= q 2 u 2 + q 2 u −1 + q 20 u 5 + q 20 u −4 + ··· .lim [θ 0 ,θ 1 ,θ 2 ](q, q −1 u)=[1, u, 0]q→0θ 0 (q, q −2 u)=1+q 3 u 3 + q 15 u −3 + q 24 u 6 + ··· ,θ 1 (q, q −2 u)=q −1 u + q 12 u −2 + q 8 u 4 + ··· ,θ 2 (q, q −2 u)=u 2 + q 3 u −1 + q 15 u 5 + q 24 u −4 + ··· ,limq→0 [θ 0,θ 1 ,θ 2 ](q, q −2 u) = limq→0[1,q −1 u, u 2 ]=[0, 1, 0] in P 2 .
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Compactification of the moduli space of abelian varieties, Kyoto ...

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