Mathematical Sciences: Algebraic Geometry in Two and Three Dimensions

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This research concerns algebraic varieties with trivial canonical bundle. They play a special role in algebraic geometry and its relations with other fields. They form the main class of examples of varieties with Kodaira number zero, which are the "building blocks" out of which many other varieties are constructed. They also play an important role in differential geometry, for example, Yau's solution of the Calabi conjecture implies that all such varieties carry a Ricci-flat Kahler- Einstein metric. Moreover in complex dimension three, varieties with such a metric have recently been of interest to physicists studying supersymmetric string theories. In this investigation the principal investigator will study several questions about algebraic varieties with Kodaira number zero, varieties with trivial canonical bundle and other related varieties. In this research the principal investigator will study the geometry of the objects one gets as the set of solutions of simultaneous polynomial equations. These objects have formed the core of geometric studies in mathematics for hundreds of years and remain a central focus today. The applications of the studies of these objects becomes ever more widespread. For example, today there are very important ramifications in physics. Morrison takes a combinatorial approach to their study and makes heavy use of the computer to determine their structure. The results of this research will be very interesting to many people.