Time：14:00-16:00, Tusday, April 21 2026
Venue：E14-301
Speaker：Gerard Freixas, École Polytechnique
Title：On the Deligne–Riemann–Roch Isomorphism
Abstract：In algebraic geometry, the Grothendieck-Riemann-Roch formula describes the functoriality of the Chern character under pushforward. In 1985, in a letter to Quillen, Deligne conjectured that the degree-one part of this formula is the shadow of a more precise statement, namely a canonical isomorphism between line bundles. This isomorphism is expected to express the Knudsen–Mumford determinant of cohomology in terms of a construction akin to an "intersection theory with values in line bundles," and to enjoy good functorial properties.
At the time, several special cases of this phenomenon were known. These include the theory of the different and discriminants, the functional equation of the Riemann theta function, certain very specific instances of Poincaré duality, and Mumford’s isomorphism for pluricanonical forms on curves. Deligne established his conjecture in the case of families of curves, using Mumford's isomorphism. Later, Elkik developed part of the theory of intersection with values in line bundles.
Recently, in joint work with Dennis Eriksson, we completed Elkik's work and constructed the isomorphism predicted by Deligne. In this talk, I will motivate the problem along the lines described above, and present our result with Dennis, together with some of its consequences.
