BoundModProbAG · Boundedness and Moduli problems in birational geometry

Boundedness and Moduli problems in birational geometry

Algebraic geometry is a sophisticated area of mathematics dating back to the mid 19th-century, that links algebra and geometry with many parts of mathematics and theoretical physics. The basic objects of study, called algebraic varieties, are the common zero sets of polynomial functions, which are higher dimensional analogs to the ellipses and hyperbolas of antiquity. The subject has key applications in very many branches of modern mathematics, science and technology. One of the main goals in algebraic geometry is to classify algebraic varieties. They can often be decomposed into simpler shapes that act as fundamental building blocks in the classification. But how many different shapes appear in each class of building blocks? Calabi-Yau varieties, characterised as flat from the point of view of curvature, are one of three types of fundamental building blocks of algebraic varieties. Calabi-Yau threefolds and fourfolds have formed the focus of interest of string theorists over recent decades. A better understanding of the geometry and the classification of Calabi-Yau varieties would advance string theory in fundamental ways, and would provide many new examples and models to study. Since they are building blocks for constructions in geometry and theoretical physics, understanding how many Calabi-Yau varieties there are is a question of fundamental importance. The problem is to know whether the shapes of Calabi-Yau varieties come in just finitely many families - a property that goes under the name of boundedness. This very difficult question remains wide open already in dimension three. Recent developments make powerful techniques available to investigate new aspects of it. This action has focused on showing that there are essentially finitely many families of Calabi-Yau varieties with some extra piece of structure -- an elliptic fibration -- in any dimension. A Calabi-Yau variety with an elliptic fibration can be decomposed like a bundle of doughnuts-like fibers over a smaller dimensional object: hence, the goal is to show that their bases are themselves bounded and then to spread the boundedness from the bases of the elliptic fibrations back to elliptic Calabi-Yau varieties.This goal was avhieved in a wide number of cases across al dimensions.

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