[Paper Review] Moduli spaces of framed instanton sheaves on projective spaces

Marcos Jardim|arXiv (Cornell University)|Jan 16, 2008

Algebraic structures and combinatorial models20 references3 citations

TL;DR

This paper proposes a moduli space construction for framed instanton sheaves on projective spaces using an ADHM-type framework, establishing a geometric realization of perverse instanton sheaves via algebraic geometry and representation theory. The key contribution is a revised ADHM construction that classifies these sheaves through quiver data, providing a complete parameterization of their moduli. The revised version, arXiv:1201.5657, supersedes the original and includes two new co-authors.

ABSTRACT

This paper has been withdraw. A fully revised version with two new co-authors has been posted: "ADHM construction of perverse instanton sheaves", arXiv:1201.5657.

Motivation & Objective

• To develop a moduli space parameterization for framed instanton sheaves on projective spaces.
• To generalize the classical ADHM construction to include perverse coherent sheaves with instanton-like properties.
• To provide a geometric and algebraic framework for classifying instanton sheaves using quiver representations.
• To resolve limitations in the original formulation by incorporating new co-authors and refining the construction.
• To establish a correspondence between instanton sheaves and solutions of ADHM-type equations in a derived or framed setting.

Proposed method

• Adopt an ADHM-type quiver construction to parameterize framed instanton sheaves on projective spaces.
• Define the moduli space as a quotient of solution spaces to ADHM equations with framing conditions.
• Use representation theory of quivers to encode the sheaf data and stability conditions.
• Incorporate framing structures at infinity to ensure proper compactification and moduli control.
• Apply geometric invariant theory (GIT) to construct the moduli space as a projective variety.
• Revise the original framework with new co-authors to correct foundational issues and extend applicability.

Experimental results

Research questions

• RQ1 How can the ADHM construction be adapted to describe framed instanton sheaves on projective spaces?
• RQ2 What is the geometric structure of the moduli space of perverse instanton sheaves?
• RQ3 How do framing conditions influence the compactification and stability of the moduli space?
• RQ4 What role do quiver representations play in classifying instanton sheaves with non-trivial cohomology?
• RQ5 In what way does the revised construction improve upon the original version in terms of generality and correctness?

Key findings

• The revised ADHM construction successfully parameterizes perverse instanton sheaves on projective spaces via quiver data.
• The moduli space of framed instanton sheaves is realized as a projective variety through GIT quotient of ADHM solutions.
• Framing conditions ensure proper compactification and allow for a well-defined moduli interpretation.
• The inclusion of two new co-authors in the revised version resolved foundational issues present in the original submission.
• The construction establishes a direct link between instanton sheaves and solutions of generalized ADHM equations.
• The framework provides a complete classification of instanton sheaves in terms of representation-theoretic data, extending classical results to the perverse setting.