Update on 3-folds
نویسنده
چکیده
The familiar division of compact Riemann surfaces into 3 cases g = 0, g = 1 and g ≥ 2 corresponds to the well known trichotomy of spherical, Euclidean and hyperbolic non-Euclidean plane geometry. Classification aims to treat all projective algebraic varieties in terms of this trichotomy; the model is Castelnuovo and Enriques’ treatment of surfaces around 1900 (reworked by Kodaira in the 1960s). The canonical class of a variety may not have a definite sign, so we usually have to beat it up before the trichotomy applies, by a minimal model program (MMP) using contractions, flips and fibre space decompositions. The classification of 3-folds was achieved by Mori and others during the 1980s. New results over the last 5 years have added many layers of subtlety to higher dimensional classification. The study of 3-folds also yields a rich crop of applications in several different branches of algebra, geometry and theoretical physics. My lecture surveys some of these topics. 1. Popular introduction: the great trichotomy A trichotomy is a logical division into three cases, where we expect to win something in each case. The cases here are similar to the “much too small, just right, much too big” of Goldilocks and the Three Bears, or the geometric division of conic sections into ellipse, parabola and hyperbola due to Appollonius of Perga (200 BC), or the cosmological question of whether the universe contracts again into a big crunch, tends to an asymptotic state or continues expanding exponentially. ∗Mathematics Institute, University of Warwick, Coventry CV4 7AL, England, UK, [email protected]
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