Deligne's example of $\deg \pi_{*}\Omega_{X/Y}<0$

Question

While reviewing Lang's book on Arakelov theory, I saw the following comment by Paul Vojta:

"...Deligne has found an example when $\deg \pi_{*}\Omega_{X/Y}$ can be negative, because Green's functions at infinity. This is of course unlike the functional field case, but this is of no consequence for next section..." (page 159)

May I ask is this example ever published? I know people are usually interested in the upper bound, not lower bound for the height functions. But I feel such a result can still be interesting to know.

Answer 1

This example should be the elliptic curve with j-invariant 0 and can be found in Deligne's paper Preuve des conjectures de Tate et de Shafarevitch (p. 29). If you are more interested in small values for elliptic curves, the article On the essential minimum of Faltings' height by Burgos Gil, Menares and Rivera-Letelier may be interesting for you. An example of negative value in genus 2 can be found in Sur le calcul explicitedes “classes de Chern” des surfaces arithmétiques de genre 2 by Bost, Mestre and Moret-Bailly.