Abelian Log Fundamental Group scheme

Abstract

Let $S$ be a connected Dedekind scheme and $X$ be a proper smooth connected scheme over $S$ . Let $D$ a divisor with no multiplicity of $X$ such that the irreducible components of $D$ and as well their intersections are smooth over $S$. Now if we endow $X$ with the log structure associated with $D$ then the structure morphism from $X$ to $S$ is log-smooth. Let $x: S \to X$ be a $S$-point such that it doesn't intersect $D$. Then we prove that the maximal abelian quotient of the log Nori fundamental group scheme of $X$ fits in to an exact sequence of the form $0 \rightarrow (\mathbf{NS}^{\tau}_{X/S,D})^{\vee} \rightarrow (\pi^\text{log}_{\text{Nori}}(X,x))^{\text{ab}} \rightarrow \underset{n}{\varprojlim} \mathbf{Alb}_{X/S,D}[n] \rightarrow 0$. Here $\mathbf{NS}^{\tau}_{X/S,D}$ is the torsion subgroup scheme of the generalized Neron-Severi group and $\mathbf{Alb}_{X/S,D}$ is the generalized Albanese scheme associated with the divisor $D$.