Non-existence of certain closed complex geodesics in the moduli space of curves (Q1104997)

Let \(A_ g\) be the moduli scheme of g-dimensional principally polarised abelian varieties and let \(M_ g\) be the moduli scheme of smooth curves of genus g. Denote by \(F: M_ g\to A_ g\) the period mapping. The author considers the maps of complete curves C of genus \(h\) into \(A_ g\) which can be factored through a map into \(M_ g\). It is proved that the image has (in the Bergman metric of \(A_ g)\) an area less than 1/3 of the expected maximum for the area of the mapping into \(A_ g\). Actually it is shown that if F(C) is a totally geodesic curve in \(A_ g\) of curvature \(-1/\ell\) then \(\ell \leq (g-1)/3\). This is proved by an application of the Bogomolov-Miyaoka-Yau inequality to the complex surface which is a restriction to C of the universal fibration over \(M_ g\). It is interesting that these arguments can be used to get some bound for \(deg(R^ 1 f^*{\mathcal O}_ X)\) of the considered fibration \(f:X\to C\). It is shown also that there are no closed geodesics of curvature -1 in \(A_ g\). From the other side in \(A_ g\) there are closed complex geodesics of curvature \(-k\) with any \(k\leq [g/2]\). Let us remark that the geodesics in \(A_ g\) parametrize families of abelian varieties with a fixed \((g-\ell)\)-dimensional abelian subvariety.