Hyper-Kahler Geometry via Lagrangian Fibrations and Symplectic Resolutions

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Algebraic Geometry has connections to many areas in mathematics, including topology, differential geometry, number theory, representation theory, combinatorics and the theory of differential equations. Over last 20 year important connections with string theory were discovered as well. Algebraic Geometry is the study of algebraic varieties: geometric objects that can be described as the collections of points satisfying a set of polynomial equations. One of the aims of the field is to classify algebraic varieties. This can be done by first associating discrete invariants to algebraic varieties and then studying all algebraic varieties with a given set of invariants. A basic invariant used in algebraic geometry, as well as in differential geometry, is the first Chern class. Algebraic varieties can be divided into classes according to the positivity properties (or lack thereof) of this invariant. One of the most important of these classes is that of varieties with first Chern class equal to zero. These varieties have a crucial role also in physics and in differential geometry. With this project the PI aims to advance our knowledge of hyper-K\"ahler manifolds which are, together with complex tori and Calabi-Yau manifolds, one of the building blocks of varieties with trivial first Chern class.More specifically, the PI plans to carry out the research in following directions: investigating the relation between hyper-Kahler manifolds and cubic fourfolds, improving the current knowledge of Lagrangian fibrations, using Lagrangian fibrations to expand our knowledge of the known examples, and carrying out a systematic study of symplectic resolutions. These lines of research build on past work of the PI as well as on recent progress in this field.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.