We consider Kobayashi geodesics in the moduli space of abelian varieties A_g
that is, algebraic curves that are totally geodesic submanifolds for the
Kobayashi metric. We show that Kobayashi geodesics can be characterized as
those curves whose logarithmic tangent bundle splits as a subbundle of the
logarithmic tangent bundle of A_g.

Both Shimura curves and Teichmueller curves are examples of Kobayashi
geodesics, but there are other examples. We show moreover that non-compact
Kobayashi geodesics always map to the locus of real multiplication and that
the $\Q$-irreducibility of the induced variation of Hodge structures implies
that they are defined over a number field.

Joint work with Martin Moeller

Lecture #13581

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