Seshadri constants, Diophantine approximation, and Roth’s theorem for arbitrary varieties

Abstract. In this paper, we associate an invariant αx(L) to an algebraic point x on an algebraic variety X with an ample line bundle L. The invariant α measures how well x can be approximated by rational points on X, with respect to the height function associated to L. We show that this invariant is closely related to the Seshadri constant ǫx(L) measuring local positivity of L at x, and in particular that Roth’s theorem on P1 generalizes as an inequality between these two invariants valid for arbitrary projective varieties. 1