[Paper Review] Homotopy groups of complements to ample divisors

Anatoly Libgober|ArXiv.org|Apr 19, 2004

Algebraic Geometry and Number Theory20 references3 citations

TL;DR

This paper investigates the homotopy groups of complements to reducible ample divisors with isolated non-normal crossings on smooth projective varieties. By generalizing vanishing theorems for fundamental group commutativity and using Hodge theory of abelian covers, it establishes that higher homotopy groups vanish in certain ranges and links their support loci to motivic zeta functions and quasiadjunction polytopes, particularly in arrangements of hyperplanes.

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

Motivation & Objective

• To extend the understanding of homotopy groups of complements to reducible divisors beyond the case of isolated singularities.
• To generalize vanishing theorems for fundamental group commutativity to higher homotopy groups.
• To characterize the support loci of non-vanishing homotopy groups as modules over the fundamental group using geometric data from non-normal crossing loci.
• To relate the motivic zeta function of Denef-Loeser to local polytopes of quasiadjunction via Hodge-theoretic invariants of abelian covers.
• To compute explicit examples, such as the second homotopy group of the complement in a Kummer configuration of planes in ℙ³.

Proposed method

• Prove that the action of the fundamental group on higher homotopy groups is trivial in certain ranges, implying homological control over these groups.
• Use homological algebra and spectral sequences to compute homotopy groups and establish vanishing in the range 2 ≤ i ≤ n−1 for divisors in ℙⁿ⁺¹ with more than n+1 components and isolated non-normal crossings.
• Define characteristic varieties as supports of non-vanishing homotopy groups in the group ring of π₁, linking them to jumping loci of local system cohomology.
• Apply Hodge theory of abelian covers to relate the Hodge realizations of motivic zeta functions to the polytopes of quasiadjunction.
• Use resolution of singularities and equivariant cohomology to compute invariants of abelian covers of links of singularities.
• Leverage the motivic zeta function in the form of a generating series over the Grothendieck ring, with Betti and Hodge realizations determining characteristic varieties and quasiadjunction data.

Experimental results

Research questions

• RQ1 Under what geometric conditions on a divisor with isolated non-normal crossings does the fundamental group act trivially on higher homotopy groups?
• RQ2 How can the support loci of non-vanishing homotopy groups be described as algebraic subvarieties of the character variety of π₁?
• RQ3 What is the precise relationship between the motivic zeta function and the polytopes of quasiadjunction in the abelian case?
• RQ4 In what range do the homotopy groups of the complement to a divisor in ℙⁿ⁺¹ vanish when the divisor has more than n+1 components and isolated non-normal crossings?
• RQ5 How can the Hodge realization of the motivic zeta function be used to compute the dimension of the F¹-filtration on the first cohomology of abelian covers?

Key findings

• For a divisor D in ℙⁿ⁺¹ with isolated non-normal crossings and more than n+1 components, πᵢ(ℙⁿ⁺¹ − D) = 0 for all i satisfying 2 ≤ i ≤ n−1.
• The action of π₁ on higher homotopy groups is trivial in the specified range, reducing the study of these groups to homological invariants.
• The support of the first non-trivial homotopy group, viewed as a π₁-module, is a characteristic variety that coincides with the jumping locus of cohomology of local systems.
• The Betti realization of the motivic zeta function determines the essential components of the characteristic variety V₁ via the limit as Tᵢ → ∞.
• For n=1, the Hodge realization of the motivic zeta function determines the polytopes of quasiadjunction, with dim F¹H¹(Xₘ₁,…,ₘᵣ)ₚ ≥ 1 if and only if the Hodge realization of the exceptional set has non-zero contribution.
• In the Kummer configuration of planes in ℙ³, the second homotopy group π₂ of the complement is non-trivial and can be computed via the methods developed.