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Boundedness and Moduli problems in birational geometry

Project description

Investigating the boundedness of the Calabi-Yau algebraic varieties

Calabi-Yau manifolds are one of the most important building blocks of algebraic varieties. Advancing understanding of the geometry and the classification of Calabi-Yau varieties would yield applications in theoretical physics as they satisfy the requirement of space for the six ‘unseen’ spatial dimensions of string theory. Investigating whether there are many families of Calabi-Yau varieties in any fixed dimension – a property called boundedness – remains a long-standing challenge. Funded by the Marie Skłodowska-Curie Actions programme, the BoundModProbAG project aims to prove that there is essentially a finite number of families of Calabi-Yau varieties with some extra piece of structure – an elliptic fibration – in any dimension.

Objective

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, called algebraic varieties, are the common zero sets of polynomial functions, which are higher dimensional analogues 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. These 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 Ricci curvature, are one of 3 types of building blocks of algebraic varieties. Calabi-Yau 3-folds and 4-folds 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 in any fixed dimension - a property that goes under the name of boundedness.

This very difficult question remains wide open. While this problem has long been considered to be out of reach, recent developments make powerful techniques available to investigate new aspects of it. The aim of this research project is to show that there is essentially a finite number of families of Calabi-Yau varieties with some extra piece of structure -- an elliptic fibration -- in any dimension.

Coordinator

ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
Net EU contribution
€ 191 149,44
Address
BATIMENT CE 3316 STATION 1
1015 Lausanne
Switzerland

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Region
Schweiz/Suisse/Svizzera Région lémanique Vaud
Activity type
Higher or Secondary Education Establishments
Links
Total cost
€ 191 149,44