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Room P3.10, Mathematics Building

Stable canonically polarized varieties I

The theory of moduli of curves has been extremely successful and part of this success is due to the compactification of the moduli space of smooth projective curves by the moduli space of stable curves. A similar construction is desirable in higher dimensions but unfortunately the methods used for curves do not produce the same results in higher dimensions. In fact, even the definition of what "stable" should mean is not clear a priori. In order to construct modular compactifications of moduli spaces of higher dimensional canonically polarized varieties one must understand the possible degenerations that would produce this desired compactification that itself is a moduli space of an enlarged class of canonically polarized varieties. In this series of lectures I will start by discussing the difficulties that arise in higher dimensions and how these lead us to the definition of stable varieties and stable families. Time permitting construction of compact moduli spaces and recent relevant results will also be discussed.