Thanks to Faltings, Arakelov and Parshin's solution to Mordell's conjecture we know that smooth complex projective curves of genus at least equal to 2 have finite number of rational points. A key input in the proof of this fundamental result is the boundedness of families of smooth projective curves of a fixed genus (greater than 1) over a fixed base scheme. The latter was generalized by the combined spectacular results of Kovacs-Lieblich and Bedulev-Viehweg to higher dimensional analogues of such curves; the so-called canonically polarized projective manifolds. In this talk I will discuss our recent extension of this boundedness result to the case of families of varieties with semiample canonical bundle (for example Calabi-Yaus). This is based on joint work with Kenneth Ascher (UC Irvine).

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This lecture is a continuation of the lecture of about a year ago with the same title. No familiarity with the last lecture will be assumed and the emphasis will be on topics that were not discussed in the previous lecture and have some novelty.