Abstract
A widely believed conjecture predicts that curves of bounded geometric genus lying on a variety of general type form a bounded family. One may even ask whether the canonical degree of a curve C in a variety of general type is bounded from above by some expression aχ(C) + b, where a and b are positive constants, with the possible exceptions corresponding to curves lying in a strict closed subset (depending on a and b). A theorem of Miyaoka proves this for smooth curves in minimal surfaces, with a > 3/2. A conjecture of Vojta claims in essence that any constant a > 1 is possible provided one restricts oneself to curves of bounded gonality. We show by explicit examples coming from the theory of Shimura varieties that in general, the constant a has to be at least equal to the dimension of the ambient variety. We also prove the desired inequality in the case of compact Shimura varieties.
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