CAREER: Hodge Theory of Calabi-Yau Varieties

About This Grant

Hodge theory is a branch of mathematics developed in the first half of the 20th century, which aims to study the non-linear geometry of shapes using linear invariants. More precisely, the shapes are complex projective manifolds and the linear invariants are Hodge structures. The famous “Hodge conjecture”, first presented in 1950 at the International Congress of Mathematicians, states that the essential geometry of a complex projective manifold is recovered by its Hodge structure. As one of the Millennium Prize Problems, this conjecture has attached to it a one million dollar prize for a solution. While complex projective manifolds involve a bit of abstraction to define, they are fundamental objects in mathematics and physics; for example, the “Calabi-Yau varieties” are a certain class of complex projective manifolds which can appear as the small dimensions of the universe in string theory. The main research goal of this project is to study the Hodge structures of Calabi-Yau varieties—to both gain insight into Hodge theory more generally for all complex projective manifolds, and to increase the depth of our understanding of Calabi-Yau varieties. Beyond pure research goals, the educational impact will be to develop an active community of young researchers and PhD students focusing on this circle of ideas. The project will continue lines of research of the PI on toroidal compactifications of Calabi-Yau moduli. Recent joint work of the PI with V. Alexeev constructs a modular toroidal compactification of the moduli space of degree 2d polarized K3 surfaces. One research goal is to extend this work to hyperkahler and Calabi-Yau varieties, with an eye towards understanding the combinatorial structure of degenerations. Additionally, the project aims to study and prove general results concerning variations of Hodge structure (VHS). Joint work of the PI with S. Tayou is advancing our understanding of finiteness properties of Z-polarized VHS, and it now seems within reach that conjectures of Deligne and Simpson on the algebraicity of the non-abelian Hodge locus can be resolved. The approach is novel, incorporating ideas from hyperbolic geometry. The PI will also pursue results about period mappings for Calabi-Yau variations of Hodge structure, such as questions of boundedness of moduli of Calabi-Yau varieties of various classes, and under what circumstances Torelli theorems hold. 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.