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 $0\to ({\mathbf{NS}}_{X/S,D}^{\tau }{)}^{\vee }\to ({\pi }_{\text{Nori}}^{\text{log}}(X,x){)}^{\text{ab}}\to \underset{n}{\underleftarrow{\lim }}{\mathbf{Alb}}_{X/S,D}[n]\to 0$. Here ${\mathbf{NS}}_{X/S,D}^{\tau }$ 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 $D$.

References in corpus (1)

• Log Fundamental Group Scheme