TITLE: "Nice" definition of discriminant as alluded to in an answer of Qing Liu
QUESTION [14 upvotes]: In his answer 
    here
 Qing Liu mentioned "the 'discriminant' of X which measures the defect of a functorial isomorphism which involves powers of the relative dualizing sheaf of X/R." 
Could somebody give a reference for or explain this?
I ask because there is a natural definition of conductor of a proper morphism to a normal scheme (for example as in Serre's "Algebraic groups and class fields"), and some relationship between the conductor and discriminant, but the definitions of discriminants I have seen always seem situation-specific and ad-hoc. 
Thanks.

REPLY [7 votes]: I want to mention that there is a published version of Deligne's letter online, namely the famous paper "Le déterminant de la cohomologie". See here:
https://publications.ias.edu/node/404
I am trying to type and translate the paper on my own, but so far I have only translated bits of it here and there as it is lengthy. I heard from Soule that the exponent $12$, is ultimately coming from the second term contribution of the expansion of the Todd class. 
In light of Brian Conrad's comment ("the oracle status of the Deligne reference is an obstruction to saying more"), I want to point out a possible mistake in Deligne's paper. Here I quote from Gerard Freixas i Montplet and Anna von Pippich's paper (see footnote of page 2):

" To be rigorous, Deligne mistakenly applies Bismut–Freed’s curvature
  theorem for Quillen connections. The subtle point is the compatibility
  of the Quillen connection with the holomorphic structure of the
  Knudsen–Mumford determinant. Nevertheless, the Riemann–Roch isometry
  he claims can be established by appealing instead to the results of
  Bismut–Gillet–Soule, where this compatibility is addressed. Notice
  however these results came later in time."

I am not sure if the issue was noticed by other people, but this may be a subtle point worth mentioning as the paper is a canonical reference in Arakelov intersection theory.