Gerard Freixas i Montplet (CNRS)

In a letter to Quillen from 1985, Deligne raised the question of lifting the degree-one part of the Grothendieck-Riemann-Roch formula to a canonical, functorial isomorphism of line bundles. This conjectural isomorphism would provide a characteristic class-type expression for the determinant of the cohomology of Knudsen-Mumford. Deligne addressed this question for families of curves, relying on previous work by Mumford and Deligne-Mumford on moduli spaces of curves. Deligne’s program remained essentially open since then. In the recent years, Dennis Eriksson and myself have been developing a functorial relative intersection theory valued in line bundles, relying on some partial developments due to Elkik in the late 80s. As a result, we have established a Grothendieck-Riemann-Roch isomorphism as conjectured by Deligne. In this talk, I will motivate and explain the statement of our result.