[Paper Review] Moduli of Admissible Pairs for Arbitrary Dimension, I: Resolution

Надежда Владимировна Тимофеева|arXiv (Cornell University)|Dec 21, 2020

Algebraic Geometry and Number Theory13 references4 citations

TL;DR

This paper introduces a higher-dimensional generalization of the standard resolution for torsion-free coherent sheaves on nonsingular projective varieties, constructing a locally free sheaf on a specially modified projective scheme. The key contribution is a modified torsion-free resolution procedure via iterated blow-ups and inductive torsion removal, enabling a compactification of moduli spaces of vector bundles that preserves the Kobayashi–Hitchin correspondence in arbitrary dimensions.

ABSTRACT

A procedure resolving a torsion-free coherent sheaf on a nonsingular $N$-dimensional projective algebraic variety into a locally free sheaf on a projective scheme of certain class is proposed. This is a higher-dimensional analog of the resolution (called the standard resolution in previous works of the author) of coherent sheaves on a surface. The method is applicable to all existing flat families of torsion-free sheaves including those who does not contain locally free sheaves. Bibliography: 23 items Keywords: moduli space, algebraic coherent sheaves, admissible pairs, vector bundles, nonsingular algebraic variety, projective algebraic variety, moduli of vector bundles, compactification of moduli.

Motivation & Objective

• To develop a higher-dimensional analog of the standard resolution for coherent sheaves on surfaces, extending it to arbitrary dimension.
• To construct a moduli scheme for semistable admissible pairs that compactifies the moduli of stable vector bundles without including non-locally free sheaves.
• To preserve the Kobayashi–Hitchin correspondence in higher dimensions by using locally free sheaves on admissible schemes.
• To provide a framework for algebraic-geometric compactification of moduli spaces of vector bundles, avoiding classical non-locally free sheaf compactifications.

Proposed method

• Proposes a resolution procedure via iterated blow-ups of a nonsingular projective variety, constructing a new projective scheme with a principal component and additional components.
• Defines a modified torsion subsheaf in the pullback of a coherent sheaf, using inductive conditions based on prime ideals of positive codimension and section extension across components.
• Constructs the quotient sheaf $\widetilde{E}_i = (\sigma_1 \circ \cdots \circ \sigma_i)^*E / \mathrm{tors}$, where $\mathrm{tors}$ is the modified torsion, ensuring the resulting sheaf is modified torsion-free.
• Uses flat morphisms $\delta_i: S_i^{\text{add}} \to D_i$ to analyze the behavior of sheaves on exceptional divisors and ensures the evaluation map $\mathrm{ev}_i^0$ is a monomorphism.
• Applies the construction inductively over a sequence of blow-ups $\sigma_i = \sigma_i^0 \circ \delta_i$, maintaining the modified torsion-free property at each step.
• Establishes that the direct image $\delta_{i*}\widetilde{E}_i$ is modified torsion-free, ensuring the resolution remains well-behaved under base change.

Experimental results

Research questions

• RQ1 How can the standard resolution of coherent sheaves on surfaces be generalized to arbitrary dimension for vector bundles on nonsingular projective varieties?
• RQ2 What is the correct notion of torsion in higher dimensions that allows for a locally free resolution without introducing non-locally free sheaves?
• RQ3 Can a compact moduli space of semistable vector bundles be constructed using locally free sheaves on admissible schemes, avoiding non-locally free coherent sheaves?
• RQ4 Does the modified torsion-free resolution preserve the Kobayashi–Hitchin correspondence in higher dimensions?
• RQ5 How can the structure of the resolution be inductively controlled through iterated blow-ups and component-wise torsion conditions?

Key findings

• The modified torsion subsheaf $\mathrm{tors}$ is defined inductively across components of the blow-up scheme, ensuring compatibility with section extension and prime ideal annihilators.
• The sheaf $\widetilde{E}_i = (\sigma_1 \circ \cdots \circ \sigma_i)^*E / \mathrm{tors}$ is modified torsion-free at each step of the resolution process.
• The evaluation map $\mathrm{ev}_i^0: \delta_{i*}\widetilde{E}_i \to \sigma^{0*}_i\widetilde{E}_{i-1}/\mathrm{tors}$ is a monomorphism, ensuring injectivity and stability of the resolution under base change.
• The direct image $\delta_{i*}\widetilde{E}_i$ is modified torsion-free, which guarantees the resolution remains well-defined and locally free in the final quotient.
• The construction yields a locally free sheaf $\widetilde{E}$ on a projective scheme $\widetilde{S}$, which is a higher-dimensional analog of the standard resolution in dimension two.
• The resolution process ensures that the resulting moduli space of admissible pairs $((\widetilde{S}, \widetilde{L}), \widetilde{E})$ is isomorphic to the Gieseker–Maruyama moduli scheme, preserving the Kobayashi–Hitchin correspondence.