[Paper Review] Notes on GIT and symplectic reduction for bundles and varieties

Richard Thomas|ArXiv.org|Dec 17, 2005

Geometry and complex manifolds35 references18 citations

TL;DR

This paper provides a geometric and intuitive introduction to Geometric Invariant Theory (GIT) and symplectic reduction, focusing on their applications to moduli spaces of vector bundles and algebraic varieties. It establishes that K-stability and slope stability coincide for smooth curves via a geometric proof based on test configurations and blow-ups, resolving a key case in K-stability theory.

ABSTRACT

These notes give an introduction to Geometric Invariant Theory and symplectic reduction, with lots of pictures and simple examples. We describe their applications to moduli of bundles and varieties, and their infinite dimensional analogues in gauge theory and the theory of special metrics on algebraic varieties. Donaldson's "quantisation" link between the infinite and finite dimensional situations is described, as are surprisingly strong connections between the bundle and variety cases.

Motivation & Objective

• To provide an accessible, geometric introduction to GIT and symplectic reduction with visual and intuitive examples.
• To unify the treatment of moduli problems for vector bundles and algebraic varieties, highlighting deep analogies between the two settings.
• To establish that K-stability and slope stability are equivalent for smooth curves using test configurations and blow-up techniques.
• To clarify the role of the Hilbert-Mumford criterion in stability analysis, particularly in infinite-dimensional analogues like cscK and KE metrics.
• To demonstrate that the flatness of thickenings in test configurations is essential for reducing stability to weight calculations on normal cones.

Proposed method

• Uses $\mathbb{C}^*$-actions and test configurations to analyze stability, particularly through deformation to the normal cone.
• Applies iterated blow-ups of $X \times \mathbb{C}$ along ideals $I = \mathcal{I}_0 + t\mathcal{I}_1 + \cdots + t^p$ to construct test configurations.
• Employs the weight of the $\mathbb{C}^*$-action on the determinant of sections of the polarized line bundle to measure stability.
• Reduces the stability condition to the sum of weights $w(Z_0) + \cdots + w(Z_{p-1}) \prec 0$, which must be negative for stability.
• Uses base change and normalization to handle non-reduced divisors, particularly by squaring the $\mathbb{C}^*$-action to simplify ideals like $(x^2, t)$.
• Applies resolution of singularities to reduce to the case of snc (simple normal crossing) divisors with multiplicity one, enabling weight comparison.

Experimental results

Research questions

• RQ1 How do GIT and symplectic reduction relate in the context of moduli spaces of bundles and varieties?
• RQ2 Can the equivalence between K-stability and slope stability for smooth curves be proven geometrically, without analytic or combinatorial methods?
• RQ3 What is the role of flatness of thickenings in the stability criterion for test configurations?
• RQ4 How can non-reduced or singular subvarieties be handled in the stability analysis using blow-ups and base change?
• RQ5 To what extent does the weight calculation on the normal cone determine the stability of a polarized variety?

Key findings

• For smooth curves, K-stability and slope stability are equivalent, and this equivalence is proven geometrically via test configurations.
• The total normalized weight of a test configuration is given by the sum $w(Z_0) + \cdots + w(Z_{p-1})$, and stability holds if and only if this sum is negative.
• The flatness of all scheme-theoretic thickenings $k\overline{(Z_i \times \mathbb{C})}$ is a sufficient condition for the weight sum to capture the full stability criterion.
• By squaring the $\mathbb{C}^*$-action and normalizing, the method resolves non-reduced structures (e.g., double points) and reduces the problem to simpler, reduced cases.
• For higher-dimensional varieties, the method suggests that slope stability is insufficient to capture full K-stability unless additional conditions on multiplicities and intersections are imposed.
• The approach via test configurations and blow-ups provides a geometric alternative to the classical combinatorial (Chow) and analytic (K-stability) proofs, offering a more intuitive framework.