On the canonical degrees of curves in varieties of general type

This paper addresses the conjecture that the canonical degree degC(KX) of a curve C in a variety X of general type is bounded from above by an expression of the form A(2gC−2)+B, where gC is the geometric genus of C and A,B are some constants, with possible exceptions corresponding to curves lying in a strictly closed subset. In particular, an effective geometric upper bound of the constants A and B in the conjecture is of interest to the authors of this paper. A theorem of Y. Miyaoka [Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 403–417; MR2426352 (2009g:14043)] proves this conjecture for smooth curves in minimal surfaces with A>3/2. A conjecture of P. Vojta [Diophantine approximations and value distribution theory, Lecture Notes in Math., 1239, Springer, Berlin, 1987; MR0883451 (91k:11049)] claims that any constant A>1 is possible provided one restricts to curves of bounded gonality. This paper shows by explicit examples coming from the theory of Shimura varieties that in general, A should be at least dimX. This paper also proves the desired inequality in the case of compact Shimura varieties which are associated to a semisimple, connected and simply connected Q-anisotropic algebraic group over Q and which contain a totally geodesic compact Riemann surface. The constructions given in the paper use the images of a totally geodesic Riemann surface under Hecke correspondences and powers of Shimura curves.