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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.
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We discuss and extend some of the results obtained in Arakelov inequalities and the uniformization of certain rigid Shimura varieties (math.AG/0503339), restricting ourselves to the two dimensional case, i.e. to surfaces Y mapping generically finite to the moduli stack of Abelian varieties. In particular we show that Y is a Hilber modular surfaces if and only if the dergee of the Hodge bundle satisfies the Arakelov equality. In the revised version, we corrected some minor mistakes, pointed out by the referee, and we tried to improve the presentation of the text.
Once first answers in any dimension to the Green-Griffiths and Kobayashi conjectures for generic algebraic hypersurfaces $mathbb{X}^{n-1} subset mathbb{P}^n(mathbb{C})$ have been reached, the principal goal is to decrease (to improve) the degree bounds, knowing that the `celestial horizon lies near $d geqslant 2n$. For Green-Griffiths algebraic degeneracy of entire holomorphic curves, we obtain: [ d ,geqslant, big(sqrt{n},{sf log},nbig)^n, ] and for Kobayashi-hyperbolicity (constancy of entire curves), we obtain: [ d ,geqslant, big(n,{sf log},nbig)^n. ] The latter improves $d geqslant n^{2n}$ obtained by Merker in arxiv.org/1807/11309/. Admitting a certain technical conjecture $I_0 geqslant widetilde{I}_0$, the method employed (Diverio-Merker-Rousseau, Berczi, Darondeau) conducts to constant power $n$, namely to: [ d ,geqslant, 2^{5n} qquad text{and, respectively, to:} qquad d ,geqslant, 4^{5n}. ] In Spring 2019, a forthcoming prepublication based on intensive computer explorations will present several subconjectures supporting the belief that $I_0 geqslant widetilde{I}_0$, a conjecture which will be established up to dimension $n = 50$.
Let $Dsubset mathbb C^n$ be a bounded domain. A pair of distinct boundary points ${p,q}$ of $D$ has the visibility property provided there exist a compact subset $K_{p,q}subset D$ and open neighborhoods $U_p$ of $p$ and $U_q$ of $q$, such that the real geodesics for the Kobayashi metric of $D$ which join points in $U_p$ and $U_q$ intersect $K_{p,q}$. Every Gromov hyperbolic convex domain enjoys the visibility property for any couple of boundary points. The Goldilocks domains introduced by Bharali and Zimmer and the log-type domains of Liu and Wang also enjoy the visibility property. In this paper we relate the growth of the Kobayashi distance near the boundary with visibility and provide new families of convex domains where that property holds. We use the same methods to provide refinements of localization results for the Kobayashi distance, and give a localized sufficient condition for visibility. We also exploit visibility to study the boundary behavior of biholomorphic maps.
In this paper we study the following slice rigidity property: given two Kobayashi complete hyperbolic manifolds $M, N$ and a collection of complex geodesics $mathcal F$ of $M$, when is it true that every holomorphic map $F:Mto N$ which maps isometrically every complex geodesic of $mathcal F$ onto a complex geodesic of $N$ is a biholomorphism? Among other things, we prove that this is the case if $M, N$ are smooth bounded strictly (linearly) convex domains, every element of $mathcal F$ contains a given point of $overline{M}$ and $mathcal F$ spans all of $M$. More general results are provided in dimension $2$ and for the unit ball.
Algebraic curves in Hilbert modular surfaces that are totally geodesic for the Kobayashi metric have very interesting geometric and arithmetic properties, e.g. they are rigid. There are very few methods known to construct such algebraic geodesics that we call Kobayashi curves. We give an explicit way of constructing Kobayashi curves using determinants of derivatives of theta functions. This construction also allows to calculate the Euler characteristics of the Teichmueller curves constructed by McMullen using Prym covers.
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