The Picard Group of the Universal Abelian Variety and the Franchetta Conjecture for Abelian Varieties

Abstract

We compute the Picard group of the universal abelian variety over the moduli stack Ag,n of principally polarized abelian varieties over C with a symplectic principal level n-structure. We then prove that over C the statement of the Franchetta conjecture holds in a suitable form for Ag,n.

The Picard group of the universal moduli stack of principal bundles on pointed smooth curves

“…where 𝐽 ∨ 𝑟 is the dual abelian stack of 𝐽 𝑟 and 𝛾 𝑟 (𝐶) is the restriction to the Neron-Severi group of 𝑟 copies of the Jacobian of some curve 𝐶. It can be proved using an argument similar to the proof of [34,Proposition 3.6]. (ii) There exists a curve of genus g ⩾ 1 over the base field 𝑘 such that the natural homomorphism id 𝐽 𝐶 ∶ ℤ → End(𝐽 𝐶 ) is an isomorphism (see [50]).…”

“… (i)There exists an exact sequence of abstract groups: $$\begin{equation} 0\rightarrow \text{Hom}_{\mathcal {M}}{\left(\mathcal {M}, J^\vee _r\right)}\xrightarrow {\nu } \operatorname{RPic}(J_r)\xrightarrow {\gamma _r(C)}\operatorname{NS}(J_C^{r}), \end{equation}$$where $$J^\vee _r$$ is the dual abelian stack of $$J_r$$ and $$\gamma _r(C)$$ is the restriction to the Neron–Severi group of $$r$$ copies of the Jacobian of some curve $$C$$. It can be proved using an argument similar to the proof of [34, Proposition 3.6]. (ii)There exists a curve of genus $$g\geqslant 1$$ over the base field $$k$$ such that the natural homomorphism $$\operatorname{id}_{J_C}:{\mathbb {Z}}\rightarrow \operatorname{End}(J_C)$$ is an isomorphism (see [50]). If $$r=1$$, then the line bundle …”

Generic cycles, Lefschetz representations, and the generalized Hodge and Bloch conjectures for Abelian varieties

“…Since the Q-subalgebra of CH * (A m ) consisting of symmetrically distinguished cycles injects in cohomology, and since Hodge classes on A m consist of polynomials in p * i L and p * i,j c 1 (P A ) for A very general (see Theorem 2.12), this would imply that generically defined cycles on m-fold powers of abelian varieties are polynomials in p * i L and p * i,j c 1 (P A ) ; see also Proposition 3.11(a) below. This would constitute a generalization (with rational coefficients) of the Franchetta conjecture for abelian varieties ; see the recent [19] where it is shown in particular that a generically defined cycle (with rational coefficients) of codimension 1 on polarized abelian varieties is a rational multiple of the polarization.…”