van der Geer: Ample on an open dense set implies non-trivial self intersection globally???

Question

Hi all

Here is something I have been stuck on for a week. In van der Geer's "The Cohomology of the Moduli Space of Abelian Varieties" he is looking at the subring of $CH_\mathbb{Q}(\tilde{\mathcal{A}_g})$ generated by the Chern classes $\lambda_i$ of the Hodge bundle $\mathbb{E}$ extended to a smooth toroidal compactification $\tilde{\mathcal{A}_g}$ of $\mathcal{A}_g$. In theorem 5.2 he writes something I have found extremely puzzling:

Since $\lambda_1$ is ample on an open dense subset $(\mathcal{A}_g)$ the socle $$\frac{1}{g(g+1)/2}(\prod ^{g}_1 (2k-1)!!) \lambda_1^{g(g+1)/2}$$ does not vanish.

For the life of me I cannot understand why $\lambda_1$ being ample on $\mathcal{A}_g$ would imply that $\lambda_1^{g(g+1)/2}$ would be non-zero. Clearly this is not true in general, with the most basic counter example being the structure sheaf of $\mathbb{P}^n$ which is ample on $\mathbb{A}^n$ because its restriction is also isomorphic to the restriction of $\mathcal{O}(1)$.

On the other hand he shows earlier that the restriction $\lambda^{g(g-1)/2}$ is non-zero on $\mathcal{A}_g$, so other than the $g=1$ case (which is easily solved regardless), $\lambda_1$ is non-trivial on $\mathcal{A}_g$. So perhaps that helps in some way, but I still do not see how to draw the conclusion.

Answer 1

Being "ample on an open set" does not mean that the restriction is ample. It means that the local conditions of being ample (separating points and tangent vectors) hold on an open set, but still using global sections of the original bundle on the original scheme. In particular, $\mathscr O_{\mathbb P^n}$ is not ample with respect to the open set $\mathbb A^n$.