Homotopy groups of complements to ample divisors

TL;DR

This paper investigates the homotopy groups of complements to ample divisors on smooth projective varieties, providing new vanishing theorems, analyzing the support of non-vanishing groups, and relating local geometric properties to global topological invariants.

Contribution

It introduces a generalized vanishing theorem for homotopy groups, connects local non-normal crossing data to global homotopy support, and relates motivic zeta functions to local polytopes of quasiadjunction.

Findings

Proved a vanishing theorem for homotopy groups of complements to ample divisors.

Described the support of non-vanishing homotopy groups in terms of non-normal crossing loci.

Linked motivic zeta functions to local geometric invariants of singularities.

Abstract

We study the homotopy groups of complements to reducible divisors on non-singular projective varieties with ample components and isolated non normal crossings. We prove a vanishing theorem generalizing conditions for commutativity of the fundamental groups. The calculation of supports of non vanishing homotopy groups as modules over the fundamental group in terms of the geometry of the locus of non-normal crossings is discussed. We review previous work on the local study of isolated non-normal crossings and relate the motivic zeta function to the local polytopes of quasiadjunction. As an application, we obtain information about the support loci of homotopy groups of arrangements of hyperplanes