Algebraic (geometric) $n$-stacks

TL;DR

This paper introduces geometric n-stacks as a generalization of algebraic stacks, using an inductive structure to explore their properties, maps from varieties, and applications to higher nonabelian cohomology and Hodge theory.

Contribution

It defines geometric n-stacks with an inductive approach, linking them to classical concepts and extending the framework for higher nonabelian cohomology and Hodge theory.

Findings

Basic properties of geometric n-stacks established

Interpretation of Brill-Noether locus as points of n-stacks

Framework for de Rham theory in higher nonabelian cohomology

Abstract

We propose a generalization of Artin's definition of algebraic stack, which we call {\em geometric $n$-stack}. The main observation is that there is an inductive structure to the definition whereby the ingredients for the definition of geometric $n$-stack involve only $n-1$-stacks and so are already previously defined. We use this inductive structure to obtain some basic properties. We look at maps from a projective variety into certain such $n$-stacks, and obtain an interpretation of the Brill-Noether locus as the set of points of a geometric $n$-stack. At the end we explain how this provides a context for looking at de Rham theory for higher nonabelian cohomology, how one can define the Hodge filtration and so on.