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Pinceaux de variétés abéliennes. (Pencils of abelian varieties). (French) Zbl 0595.14032
Astérisque, 129. Publié avec le concours du Centre National de la Recherche Scientifique. Paris: Société Mathématique de France. 266 p. FF 155.00; \$ 18.00 (1985).
Denote by $${\mathcal A}={\mathcal A}_{g,d,n}$$ the (fine) moduli space of abelian varieties of dimension g, polarized of degree $$d^ 2$$ with $$level\quad n$$ structure (n$$\geq 3)$$. It is known that $${\mathcal A}$$ is a quasiprojective scheme (of finite type) over Spec($${\mathbb{Z}}[1/n])$$. With the universal family of abelian varieties $$f:\quad {\mathcal X}\to {\mathcal A}$$ we have the invertible sheaf $${\bar \omega}=f_*\Omega^ g_{{\mathcal X}/{\mathcal A}}$$. - One of the main theorems of this article is: The sheaf $${\bar \omega}$$ is ample on $${\mathcal A}.$$
For the proof the author constructs a totally symmetric (notion defined by the author; to say roughly, symmetric with cubic structure) invertible sheaf $${\mathcal L}$$ on $${\mathcal X}$$ whose polarization is 4 times the canonical one and shows the following two statements: $$(A)\quad \det (f_*{\mathcal L})^{\vee}$$ is ample with respect to f; $$(B)\quad (\det f_*{\mathcal L})^{\otimes 2}\otimes {\bar \omega}^{\otimes 4^ gd}$$ is a torsion in Pic($${\mathcal A}).$$
In the classical case (over $${\mathbb{C}})$$ such procedure (and the above theorem) corresponds to main results of the theory of theta functions: for every ample line bundle the sections are given by theta functions; the theta zero-values generate essentially the sheaf of automorphic forms ($${\bar \omega}$$) which is shown to be ample.
The statement (B) is a result of a more general one: Let $$f:\quad {\mathcal A}\to S$$ be a family of g-dimensional abelian schemes over a scheme S and $${\mathcal L}^ a$$symmetric invertible sheaf on $${\mathcal A}$$ which is ample with respect to S of degree $$d^ 2$$. We consider $${\bar \omega}{}_{{\mathcal A}/S}=f_*\Omega^ g_{{\mathcal A}/S}$$. Then $$(\bigwedge^ df_*L)^{\otimes 2}\otimes {\bar \omega}^{\otimes d}_{{\mathcal A}/S}$$ is a torsion in Pic(S). - The author proves this claim when d is invertible on S or S is normal and excellent. He then tries to extend the results in the case when the abelian varieties have only stable reductions. Let us consider such a family $$f:\quad {\mathcal A}\to S$$ with ample symmetric sheaf $${\mathcal L}_{\eta}$$ over the generic point. For that purpose the author first generalizes the notion of cubic structure in such case and proves that a cubic structure on the generic fibre extends uniquely to a cubic $${\mathcal L}$$ over S (after a finite base change). We assume here that, by denoting K($${\mathcal L})$$ the kernel of the associated polarization $$\phi_{{\mathcal L}}:\quad {\mathcal A}\to {\mathcal A}^ t,$$ K($${\mathcal L})$$ is finite over S. The author proves the following under a slightly stronger assumption that K($${\mathcal L}^{\otimes 2})$$ is finite over S which is normal, noetherian and excellent: $$(\bigwedge^ df_*{\mathcal L})^{\otimes 2}\otimes {\bar \omega}^{\otimes d}_{{\mathcal A}/S}$$ is torsion in Pic(S).- From this one can show: Theorem. Under the assumption above, for some $$n\leq 1$$, $${\bar \omega}{}^{\otimes d}_{{\mathcal A}/S}$$ is generated by sections.
As an application the author reproves a theorem of Zarkhin on the finiteness of abelian varieties with given isogeny class over a function field over a finite field, which was according to the author one of the motivations of his study.
Reviewer: Y.Namikawa

##### MSC:
 14K10 Algebraic moduli of abelian varieties, classification 14K05 Algebraic theory of abelian varieties 14K15 Arithmetic ground fields for abelian varieties