Proposition 1. 6.3.2. The pullback of P by (xx, jv,−1) : C ×S J → J ×S J∨ together with its rigidifications at b and 0, is equal to Luniv. J/S Let d be in Z≥0. The morphism : C(d) → J∨ = Pic0 is the composition of first Σ: C(d) → J, sending, for every S-scheme T, each point D in C(d)(T ) to the class of OCT (D − db) twisted by the pullback from T that makes it rigidified at b, followed by jv,−1 : J → J∨. Summarised in a diagram, with M := (id × jv,−1)vP: b Luniv P M (1.6.3.3) b i˜d×Σ Nd jb×j𝗁,−1 id×j𝗁,−1 id×Σ ( ) C ×S J J ×S J∨ J ×S J J ×S C d .
Proposition 1. 6.7.1. The invertible O-module M on (J ×Zq J)Qq , with its rigidifica- tions, extends uniquely to an invertible O-module M with rigidifications on J ×Zq J . The biextension structure on M× extends uniquely to a biextension structure on M˜×.
Proposition 1. Under the take-it-or-leave-it offer, if the condition, u0(Γj+1) + uj(Γj+1) > πj u0(Γj,G) + uj(Γj,G) + (1 − πj) u0(Γj,B) + uj(Γj,B) − c0 − cj (3) holds, then the brand B and the generic Gj, agree in Γj on the P4D payment,
Proposition 1. 3.1. (Serre [37]) Two divisors X1 and X2 are alge- braically equivalent if and only if ϕX1 = ϕX2 . →
Proposition 1. 4.2. (Birkenhake–Xxxxx [6, Proposition 13.3.1] and Xxxx [20, Theorem 1.4.1-(iii)]) With the notation above, the complex torus AΦ,m is an abelian variety and has a natural CM structure given by the action of OK on m. In this thesis, we use the complex torus Cg/Φ˜(m) instead of AΦ,m as a realization of an abelian variety over C conforming to the notation of Lang [20] and Xxxxxxx–Taniyama [40]. For each α ∈ K, we let SΦ(α) be the matrix diag(φ1(α), . . . , φg (α)). Theorem 1.4.3. (Lang [20, Theorem 1.4.1-(ii)]) Let (K, Φ) be a CM pair and let (A, θ) be an abelian variety of type (K, Φ) with CM by K. Then there is a fractional ideal m I and an analytic isomorphism ι : Cg/Φ˜(m) → A(C) such that the diagram Cg/Φ˜(m) A(C) SΦ(α) θ(α) Cg/Φ˜(m) A(C) commutes for all α ∈ OK. Definition 1.4.4. We say that an abelian variety (A, θ) of type (K, Φ) is of type (K, Φ, m) if there is a fractional ideal m I and an analytic isomorphism ι : Cg/Φ˜(m) → A(C). Definition 1.4.5. Let (A, θ) be an abelian variety of type (K, Φ) and let X be an ample divisor on A. We say that (A, θ) is Φ-admissible with respect to the polarization ϕX if θ(K) is stable under the Xxxxxx involution. Theorem 1.4.6. (Lang [20, Theorem 1.4.5-(iii)] If an abelian variety (A, θ) of type (K, Φ, m) is simple, then it is Φ-admissible with respect to every polarization.
Proposition 1. Let (A1, θ1) and (A2, θ2) be abelian varieties of prim- itive CM type (K, Φ). If an isogeny λ from (A1, θ1) onto (A2, θ2) is an a- multiplication, then λ∗ from (A∗1, θ1∗) onto (A∗2, θ2∗) is an a-multiplication.
Proposition 1. .4.1. Let M be an affine log Xxxxxx-Xxx surface with maximal bound- ary, then M admits a cyclic dilation if and only if it admits a quasi-dilation.
Proposition 1. 1.1. For each a in X, there exists a unique real-valued ga in C∞(X − {a}) such that the following properties hold:
Proposition 1. 3.2. 1. P(a0, . . . , an) ∼= P(qa0, . . . , qan) for any q ∈ N; ^
Proposition 1. It is not the case that τ (T ) ∼= P3 with its middle vertex corresponding to a degree 4 vertex of T.
