Degenerations of Kahler manifolds and special non-compact Kahler metrics

Complex manifolds are higher-dimensional geometric spaces defined using the complex numbers. This project focuses on two special kinds of such spaces, namely Calabi-Yau manifolds, and Kähler-Ricci solitons. These types of complex manifolds have a wide array of applications throughout physics and mathematics, and for this reason have attracted a great deal of attention in recent years. This project will focus on constructions and classifications of such complex manifolds. The results will deepen our understanding of the geometry of complex manifolds as a whole and more broadly will further advance our knowledge through connections to other subjects ranging from algebraic geometry, topology, and analysis to physics. The project will also be used to help support researchers interested in contributing to these active and competitive fields. More technically, the project aims to further our understanding of the singularity development both of the Kähler-Ricci flow and of collapsing Calabi-Yau manifolds. Characterizing the development of singularities along the Kähler-Ricci flow is one of the fundamental questions in geometric analysis and is expected to reveal deep ties between geometric analysis, algebraic geometry, and topology. On the other hand, the analogous collapsing Calabi-Yau picture has many applications, including to physics. These two programs fit into a common framework, each with a given singularity model (respectively Kähler-Ricci solitons and complete non-compact Calabi-Yau metrics) and a corresponding type of degeneration (respectively the Kähler-Ricci flow and families of collapsing Calabi-Yau manifolds). The project will (1) construct examples of non-compact singularity model geometries, (2) describe the moduli of these spaces and (3) analyze the corresponding degenerations. 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.