Hyperkähler Geometry, Stability Conditions, and Moduli Spaces

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Algebraic geometry studies the solution sets of systems of polynomial equations. A theme of the subject is to classify different geometric shapes that naturally arise in the study of these solution sets. This has a root in classical problems, and nowadays it finds deep connections with modern tools, especially moduli spaces, stability conditions, and derived categories. This project aims at further applications of these tools to concrete geometric questions open for a long time. It also supports graduate students to explore the subject through travel opportunities to conferences and workshops.This project contains three related research goals: The first is a systematic study of the connection between Fano geometry and hyperkähler geometry, like generalized Kummer varieties and Gushel-Mukai varieties, using tools from derived categories. The second goal is to construct stability conditions in several important cases including Calabi-Yau threefolds, Fano fourfolds, and hyperkähler varieties. The main approach is via various restriction theorems to reduce to lower dimensional cases. The third goal is to revisit several long-standing questions on moduli of sheaves on surfaces, including the computation of Picard groups and linear series. The recent development in moduli theory will play a central role in this study.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.