[Paper Review] Stable pairs on curves and surfaces

Daniel Huybrechts, Manfred Lehn|ArXiv.org|Nov 9, 1992

Algebraic Geometry and Number TheoryMathematics8 references100 citations

TL;DR

This paper establishes the existence of fine quasi-projective moduli spaces for stable pairs $({\cal E}, \alpha: {\cal E} \to {\cal E}_0)$ on smooth projective curves and surfaces using geometric invariant theory, introducing a stability condition parameterized by a polynomial $\delta$. The key contribution is a compactification of moduli spaces for framed bundles and Higgs pairs, generalizing previous results on algebraic spaces to quasi-projective schemes.

ABSTRACT

We describe stability conditions for pairs consisting of a coherent sheaf and a homomorphism to a fixed coherent sheaf on a projective variety. The corresponding moduli spaces are constructed for pairs on curves and surfaces. We consider two examples. The fixed sheaf is the structure sheaf or is a vector bundle on a divisor, i.e. Higgs pairs or framed bundles, resp. (unencoded version)

Motivation & Objective

• To construct fine quasi-projective moduli spaces for stable pairs $({\cal E}, \alpha: {\cal E} \to {\cal E}_0)$ on smooth projective curves and surfaces.
• To generalize the notion of stability for pairs by introducing a parameter $\delta$, extending classical stability for vector bundles.
• To provide a compactification of moduli spaces for framed bundles and Higgs pairs, which were previously only constructed as algebraic spaces.
• To establish a connection between framed bundles, level structures, and Higgs pairs via dualization and stability conditions.
• To generalize Bogomolov's restriction theorem to stable pairs on high-degree curves in surfaces.

Proposed method

• Define stability for pairs $({\cal E}, \alpha)$ with respect to a polynomial $\delta > 0$ via two inequalities involving Hilbert polynomials and ranks of subsheaves.
• Use geometric invariant theory (GIT) to construct projective moduli spaces for stable pairs on curves and surfaces.
• Employ boundedness results and sectional stability to ensure the existence of fine moduli spaces.
• Analyze the moduli construction via invariant theory, proving technical Proposition 1.18 to ensure properness and separatedness.
• Dualize the Higgs pair construction to obtain framed bundles with a homomorphism to $\mathcal{O}_X$, enabling compactification via torsion-free sheaves.
• Apply the stability parameter $\delta$ to approximate the moduli space of semistable bundles in the limit, generalizing Thaddeus' method for surfaces.

Experimental results

Research questions

• RQ1 Can a quasi-projective moduli space be constructed for stable pairs $({\cal E}, \alpha: {\cal E} \to {\cal E}_0)$ on curves and surfaces?
• RQ2 How does the stability condition for pairs depend on the additional parameter $\delta$, and what is its geometric interpretation?
• RQ3 Can the moduli space of framed bundles be compactified and shown to be quasi-projective using this framework?
• RQ4 How do Higgs pairs and their duals relate to framed bundles and level structures in the context of moduli spaces?
• RQ5 What is the behavior of stable pairs under restriction to high-degree curves, and does a Bogomolov-type inequality hold?

Key findings

• For smooth projective curves and surfaces, there exists a fine quasi-projective moduli space of stable pairs $({\cal E}, \alpha)$ with respect to a polynomial $\delta$, as stated in Theorem 1.21.
• The moduli space can be naturally compactified by including pairs with torsion-free sheaves, extending the construction beyond locally free sheaves.
• When ${\cal E}_0 \cong \mathcal{O}_X$, the stable pairs correspond to Higgs pairs, and the moduli space of rank two Higgs pairs on a curve is realized as a quasi-projective scheme.
• For framed bundles with ${\cal E}_0 \cong \mathcal{O}_C^{\oplus r}$, the moduli space is quasi-projective under the condition $\delta_1 < (r-1)(C.H)$, generalizing earlier results.
• The condition $\max_{0<s<r} \left\{ \frac{r s}{r-s} \sum a_i \nu_s({\cal E}_0, C_i) \right\} < \delta_1 < (r-1)(C.H)$ ensures $\mu$-stability of framed bundles, proving quasi-projectivity of the moduli space.
• The paper generalizes Bogomolov’s restriction theorem to stable pairs, showing that stable pairs restrict to stable bundles on curves of sufficiently high degree.