Kummer Extensions: Applications to Generalized Cross-Ratios and Functions on Hilbert Schemes

Explore Kummer extensions and their applications in a lecture by Spencer Bloch from the University of Chicago. Delve into the geometric and arithmetic constructions of these extensions of Hodge structure. Learn how to create Kummer extensions in two scenarios: one involving smooth projective varieties over complex numbers with algebraic cycles, and another based on motivic cohomology classes and algebraic cycle classes. Discover the concept of generalized cross-ratios and their relation to functions on Hilbert schemes. Examine the connection between Kummer extensions and the work of A. Mellit on special cases of the Gross-Zagier theorem. Gain insights into the interplay between algebraic geometry, arithmetic, and Hodge theory in this advanced mathematical exploration.