[Paper Review] Moduli of vector-bundles on surfaces

Kieran G. O’Grady|ArXiv.org|Sep 20, 1996

Algebraic Geometry and Number Theory27 references15 citations

TL;DR

This paper establishes conditions under which the moduli space of H-semistable torsion-free sheaves on a smooth complex projective surface is reduced and of the expected dimension, with irreducibility and density of stable bundles for large discriminant Δ. The results extend to arbitrary rank and rely on algebro-geometric techniques inspired by Donaldson theory, including deformation theory and holomorphic two-forms on moduli spaces.

ABSTRACT

This is a survey paper: we discuss certain recent results, with some improvements. It will appear in the S. Cruz proceedings.

Motivation & Objective

• To prove that the moduli space of H-semistable torsion-free sheaves on a smooth complex projective surface is reduced and of the expected dimension when the discriminant Δ exceeds a rank-dependent threshold Δ(r).
• To show that for large Δ, the open subspace parametrizing slope-stable vector bundles is dense and the full moduli space is irreducible.
• To extend results on holomorphic two-forms and Kodaira dimension of the moduli space to arbitrary rank under a conjecture on vector bundles over curves.
• To provide a framework for understanding moduli spaces on surfaces of general type, where little was previously known.
• To establish conditions under which the moduli space is a fine moduli space via a gcd condition on rank, first Chern class, and Euler characteristic.

Proposed method

• Use deformation theory to compute the expected dimension of the moduli space via the Euler characteristic of the Ext complex: $ \chi(\text{End}(F)) = 2r c_2 - (r-1)c_1^2 - (r^2-1)\chi(\mathcal{O}_S) + h^1(\mathcal{O}_S) $.
• Apply the Gieseker-Maruyama theory of semistable sheaves, using Jordan-Hölder filtrations and S-equivalence to construct a projective moduli space.
• Employ twisted endomorphisms and cohomological techniques to analyze the tangent space and obstruction theory of the moduli space.
• Leverage Mukai's construction to lift holomorphic two-forms ω on S to two-forms ω_ξ on the moduli space, analyzing their non-degeneracy.
• Use J. Li's result on the Kodaira dimension of moduli spaces for surfaces of general type, under conditions on rank and Chern classes.
• Verify a key conjecture on vector bundles over curves (Conjecture 2.4) for arbitrary rank and special degree (Proposition 2.5), enabling extension of results to higher rank.

Experimental results

Research questions

• RQ1 Under what conditions is the moduli space of H-semistable torsion-free sheaves on a surface reduced and of the expected dimension?
• RQ2 How does the moduli space behave for large discriminant Δ, particularly regarding irreducibility and density of stable bundles?
• RQ3 When is the holomorphic two-form ω_ξ on the moduli space non-degenerate, and what does this imply for the geometry of the moduli space?
• RQ4 What is the Kodaira dimension of the moduli space when the surface S is of general type and the rank is two?
• RQ5 To what extent can results on two-forms and Kodaira dimension be extended to arbitrary rank, assuming a conjecture on vector bundles over curves?

Key findings

• For any smooth complex projective surface S and ample divisor H, the moduli space $ \mathcal{M}_\xi $ of H-semistable torsion-free sheaves with fixed rank r, determinant det(ξ), and c₂(ξ) is reduced and of the expected dimension when Δ_ξ > Δ(r).
• For Δ_ξ ≫ 0, the open subspace parametrizing slope-stable vector bundles is dense in $ \mathcal{M}_\xi $, and $ \mathcal{M}_\xi $ is irreducible.
• The holomorphic two-form ω_ξ on $ \mathcal{M}_\xi $, induced from a holomorphic two-form ω on S, is non-degenerate at the generic point when r=2 and under additional hypotheses.
• The image of the map $ \mathcal{M}_\xi \to CH_0(S) $ sending a sheaf to its c₂-class has dimension equal to that of $ \mathcal{M}_\xi $, due to non-degeneracy of ω_ξ.
• If S is of general type and r=2 with suitable conditions, the moduli space $ \mathcal{M}_\xi $ is of general type, as shown by J. Li.
• The results on non-degeneracy of ω_ξ and Kodaira dimension extend to arbitrary rank under the assumption of Conjecture 2.4, which is verified for arbitrary rank and a special degree in Proposition 2.5.