Kobayashi geodesics in \(\mathcal A_g\). (English) Zbl 1218.14036

Let \({\mathcal A}_g\) denote a moduli space of polarized Abelian varieties with toroidal compactification \(\overline{\mathcal A}_g\) and let \(Y_0\) be an open dense subscheme of a complex smooth projective curve \(Y\). A polarized \(\mathbb{Z}\)-variation of Hodge structures on \(Y_0\) gives rise to a morphism \(\varphi_0: Y_0\to{\mathcal A}_g\).
The main result of the paper is the following characterization of those curves in \({\mathcal A}_g\) which are Kobayashi geodesics, i.e. totally geodesic submanifold for the Kobayashi metric: Denote \(S= Y\setminus Y_0\) and \(\varphi: Y\to\overline{\mathcal A}_g\) the extension of \(\varphi_0\). Then \(\varphi_0: Y_0\to{\mathcal A}_g\) is a Kobayashi geodesic if and only if the natural map \(\varphi^*\Omega{1\over{\mathcal A}_g}(\log S_{\overline{\mathcal A}_g})\to \Omega^1_Y(\log S)\) splits, and this is the case if and only if the variation of Hodge structures contains a non-unitary subvariation of Hodge structures which satisfies the Arakelov equality as defined in the paper. It is shown that Shimura and Teichmüller curves are Kobayashi geodesics, but there are other examples. Moreover, noncompact Kobayashi geodesics always map to the locus of real multiplication, and irreducibility over \(\mathbb{Q}\) of the induced variation of Hodge structures implies that they are defined over a number field.


14K12 Subvarieties of abelian varieties
14G35 Modular and Shimura varieties
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