Abelian Varieties, Jacobians, and Applications

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The PI proposes to work with multiple collaborators to study topics in algebraic geometry, number theory, and string theory related to abelian varieties, curves, and their moduli. The PI will study the intersection homology of the moduli spaces of abelian varieties, investigate the failure of injectivity of the Torelli map for Prym varieties, and endeavor to prove by degeneration that the Schottky-Jung identities characterize Jacobians of curves among all abelian varieties. The PI will attempt to use the geometric properties of Jacobians to approach Coleman's conjecture on the finiteness of the number of Jacobians of a fixed large genus with complex multiplication. By using real-normalized meromorphic differentials, the PI will work on constructing complete subvarieties of the moduli space of curves. The PI will also investigate further mathematical and physical properties of superstring scattering amplitudes an ansatz for which he proposed.Algebraic curves (aka Riemann surfaces) are real two-dimensional surfaces with a metric on them. They arise in many areas of mathematics, and are also fundamental to string theory as worldsheets (trajectories) of strings propagating in space. One can associate to any algebraic curve its Jacobian - it is an algebraic variety (a geometric set of points, such that there is an operation of "adding" two points together), and knowing the Jacobian one can recover the curve uniquely. This project aims to utilize and further study the intricate interplay between the geometry of the curve and of its Jacobian in order to further our understanding of curves and abelian varieties. Progress made on the questions addressed by this project would have implications in mathematics and physics going beyond algebraic and complex geometry, particularly in number theory, integrable systems, partial differential equations, and perturbative string theory.