Subvarieties of moduli stacks.2005 AMS Summer Institute on Algebraic GeometrySándor Kovács, University of Washington At the 1962 ICM Shafarevich announced a conjecture regarding finiteness properties of families of smooth projective curves. It was confirmed in the geometric case by Parshin (1968) and Arakelov (1971), and in the arithmetic case by Faltings (1983). This conjecture is related to many other problems, perhaps the most famous one is the Mordell conjecture: by a very nice argument, now known as "Parshin's covering trick" the Mordell conjecture follows from the Shafarevich conjecture. During the past 10 years years many results have been obtained with regard to higher dimensional generalizations of Shafarevich's conjecture in the geometric case. In this talk I will review the original conjecture, it's possible generalizations, and the current knowledge about hyperbolicity, boundedness and rigidity of families of varieties of non-negative Kodaira dimension. Among others I will review the work of Viehweg and Zuo as well as my own contributions.
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