CAREER: Compact Hyper-Kahler manifolds and Lagrangian fibrations

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Algebraic geometry is the study of algebraic varieties, geometric objects described by polynomial equations. To study algebraic varieties, it is often convenient to divide them into different classes according to their geometric properties. An important class of algebraic varieties is given by those whose first Chern class, a basic invariant of algebraic varieties, is zero. A fundamental result from the 1980s, called the Beauville-Bogomolov theorem, states that there are exactly three kinds of building blocks for smooth compact algebraic varieties with zero first Chern class: complex tori, strict Calabi-Yau manifolds, and irreducible holomorphic symplectic manifolds. This project focuses on the last of these three building blocks, which traditionally has been the least studied. Thanks to some fundamental theorems in differential geometry, irreducible holomorphic symplectic manifolds admit a special metric called a hyper-Kähler metric. The geometry of holomorphic symplectic manifolds is relevant not only to algebraic and differential geometry, but also to representation theory and mathematical physics. As part of this project, the PI will organize activities to strengthen the hyper-Kähler research community in the United States, activities for undergraduate students from under-represented minorities in math, and K-12 activities in the broader community around Columbia University.K3 surfaces constitute one of the most studied types of algebraic surfaces, and irreducible symplectic manifolds are arguably their higher dimensional analogues. In this analogy, Lagrangian fibrations on compact hyper-Kähler manifolds are the natural generalizations of elliptic K3 surfaces. Together with symplectic resolutions, Lagrangian fibrations provide one the strongest means to study, classify, and construct this class of manifolds. The PI aims to advance the current knowledge of (compact) hyper-Kähler manifolds through the systematic study of Lagrangian fibrations. More specifically, the PI will introduce new techniques to compactify quasi-projective Lagrangian fibrations and will study the cohomology, derived categories, and Chow groups of Lagrangian fibered compact hyper-Kähler manifolds.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.