TITLE: "Albanese" schemes: When does an "initial abelian scheme" exist under a given scheme? 
QUESTION [13 upvotes]: For a variety V, its Albenese variety Alb(V) is a variety with a map V → Alb(V) which factors uniquely into any map from V to an abelian variety.  Can we say something similar for an arbitrary scheme?  When do we know there exists an "Albanese" scheme Alb(X)?  That is,

Under what conditions on a scheme X does there exist a morphism X → Alb(X) which factors uniquely into any map from from X to an abelian scheme?

REPLY [2 votes]: The recent paper "Para-abelian varieties and Albanese maps" by Bruno Laurent and Stefan Schröer (https://arxiv.org/abs/2101.10829) revived the topic and seems to give an almost optimal existence result which I'll leave you discover in the article itself.