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. 3.2. 1. P(a0, . . . , an) ∼= P(qa0, . . . , qan) for any q ∈ N; ^
Proposition 1. Under take-it-or-leave-it offer, if the condition, j j j j
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. It is not the case that τ (T ) ∼= P3 with its middle vertex corresponding to a degree 4 vertex of T.
Proposition 1. The category Conv = ( ) of Xxxxxxxxx–Xxxxx algebras is both complete and cocomplete, and it is symmetric monoidal closed. The tensor unit is the final xxxxxxxxx set 1, since D(1) ∼= 1. the form γ : D(X)γ→ X sati sfying γ '◦ η = id and γ ◦ μ = γ ◦ D(γ). A We recall that an Xxxxxxxxx–Xxxxx algebra (of the monad D) is a map of D → ◦ ◦ D morphism D(X) → X −→ D(X′) γ X′ in EM(D) is a map f : X → X′ with f γ = γ′ (f ). An import→ant point is that we identify an algebra with a convex set: the map γ : (X) X turns a formal convex combination into an actual element in X. Maps of algebras preserve such convex sums and are commonly called affine functions. Therefore we often write Conv for the category EM(D).
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)). ∈ K O 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. ∈ K 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. Chapter 1. Preliminaries 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.
