[Paper Review] Lefschetz Hyperplane Theorem for Stacks

Daniel Halpern-Leistner|arXiv (Cornell University)|Aug 4, 2010

Homotopy and Cohomology in Algebraic Topology2 references3 citations

TL;DR

This paper establishes the Lefschetz Hyperplane Theorem for compact smooth Deligne-Mumford stacks over complex manifolds using Morse theory and Łojasiewicz's inequality, showing that a stack deformation retracts onto its hyperplane section via attachments of high-dimensional finite disc quotients. The key contribution is proving the theorem without nondegeneracy or genericity assumptions on the hyperplane section, extending classical results to stacks with integral coefficients and homotopy groups.

ABSTRACT

We use Morse theory to prove that the Lefschetz Hyperplane Theorem holds for compact smooth Deligne-Mumford stacks over the site of complex manifolds. For $Z \subset X$ a hyperplane section, $X$ can be obtained from $Z$ by a sequence of deformation retracts and attachments of high-dimensional finite disc quotients. We use this to derive more familiar statements about the relative homotopy, homology, and cohomology groups of the pair $(X,Z)$. We also prove some preliminary results suggesting that the Lefschetz Hyperplane Theorem holds for Artin stacks as well. One technical innovation is to reintroduce an inequality of Łojasiewicz which allows us to prove the theorem without any genericity or nondegeneracy hypotheses on $Z$.

Motivation & Objective

• To extend the classical Lefschetz Hyperplane Theorem to compact smooth Deligne-Mumford stacks over complex manifolds.
• To establish the theorem with integral coefficients and in terms of homotopy groups, not just rational homology.
• To remove the need for nondegeneracy or genericity conditions on the hyperplane section using Łojasiewicz's inequality.
• To develop Morse theory for stacks in a 2-categorical framework, proving cell decompositions via 2-colimits.
• To explore the possibility of extending the result to smooth Artin stacks.

Proposed method

• Adapts Bott’s Morse-theoretic proof of the Lefschetz Hyperplane Theorem to the setting of Deligne-Mumford stacks.
• Applies Łojasiewicz’s inequality to ensure stable gradient flow even without nondegeneracy of the section.
• Uses the Łojasiewicz inequality (L) to prove that the negative gradient flow of a real analytic function extends to a deformation retract onto the zero locus.
• Constructs cell attachments as 2-categorical colimits along embedded substacks, ensuring the decomposition is valid at the stack level.
• Applies the theory to show that the stack X is homotopy equivalent to Z with high-dimensional disc quotients attached.
• Leverages Hepworth’s foundational work on Morse theory for the underlying space of DM stacks, extending it to the stack level.

Experimental results

Research questions

• RQ1 Can the Lefschetz Hyperplane Theorem be extended to smooth Deligne-Mumford stacks with integral coefficients and homotopy groups, not just rational cohomology?
• RQ2 Does the Lefschetz Hyperplane Theorem hold for stacks without requiring nondegeneracy or genericity of the hyperplane section?
• RQ3 Can Morse theory for stacks be formulated in a 2-categorical sense, with cell attachments as 2-colimits?
• RQ4 Do the results extend to smooth Artin stacks, particularly those with compact presentations?
• RQ5 What role does Łojasiewicz’s inequality play in stabilizing gradient flows for singular or degenerate functions in this context?

Key findings

• The Lefschetz Hyperplane Theorem holds for compact smooth Deligne-Mumford stacks with integral coefficients, not just rationally.
• The stack X deformation retracts onto its hyperplane section Z via a sequence of deformation retracts and attachments of high-dimensional finite disc quotients.
• The theorem holds without any nondegeneracy or genericity assumptions on the section s, thanks to Łojasiewicz’s inequality.
• The negative gradient flow of a real analytic function f on X extends to a continuous deformation retraction onto the zero locus Z, provided the Łojasiewicz inequality (L) holds.
• The Łojasiewicz inequality is satisfied by functions that are real analytic up to a nonvanishing smooth factor, which includes many natural geometric functions.
• Preliminary evidence suggests the homological Lefschetz theorem may extend to a broad class of smooth Artin stacks, particularly those with compact presentations.