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. 



Deligne's example of $\deg \pi_{*}\Omega_{X/Y}<0$
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](http://www.numdam.org/article/SB_1983-1984__26__25_0.pdf) (p. 29). If you are more interested in small values for elliptic curves, the article [On the essential minimum of Faltings' height](https://arxiv.org/pdf/1609.00071.pdf) 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](http://www.numdam.org/article/AST_1990__183__69_0.pdf) by Bost, Mestre and Moret-Bailly.


