[Paper Review] Automorphisms of moduli spaces of vector bundles over a curve

Indranil Biswas, Tomás L. Gómez+1 more|arXiv (Cornell University)|Feb 14, 2012

Algebraic Geometry and Number Theory7 references5 citations

TL;DR

This paper provides a simplified proof of the automorphism group structure of moduli spaces of stable vector bundles on a smooth complex projective curve of genus $ g \geq 4 $. It shows that automorphisms arise from curve automorphisms, tensorization with line bundles, and, when $ r \mid 2\deg(\Lambda) $, dualization; the proof uses the Hitchin discriminant and nilpotent cone bundles, avoiding Hecke transforms and minimal rational curves.

ABSTRACT

Let X be an irreducible smooth complex projective curve of genus g at least 4. Let M(r,Λ) be the moduli space of stable vector bundles over X or rank r and fixed determinant Λ, of degree d. We give a new proof of the fact that the automorphism group of M(r,Λ) is generated by automorphisms of the curve X, tensorization with suitable line bundles, and, if r divides 2d, also dualization of vector bundles.

Motivation & Objective

• To provide a streamlined proof of the automorphism group structure of $ M(r,\Lambda) $, the moduli space of stable vector bundles of rank $ r $ and fixed determinant $ \Lambda $ on a curve $ X $ of genus $ g \geq 4 $.
• To eliminate reliance on complex tools such as Hecke transforms and minimal rational curves used in prior proofs.
• To reconstruct automorphisms directly from the geometry of the Hitchin discriminant and nilpotent cone bundles.
• To extend the result uniformly to all $ g \geq 4 $, avoiding special treatment for low genus cases.
• To establish a Torelli-type theorem for $ M(r,\Lambda) $, showing that the moduli space determines the curve and rank.

Proposed method

• Use the Hitchin map and its discriminant (locus of singular spectral curves) to analyze the geometry of the moduli space.
• Reconstruct the nilpotent cone bundles from the Hitchin discriminant, which encode the automorphism data of generic stable bundles.
• Apply the Lie algebra structure of $ \operatorname{End}_0 E $ to classify possible automorphisms via reductions of structure groups in $ \operatorname{PGL}_r $-bundles.
• Use the short exact sequence $ 1 \to \mathbb{C}^* \to \operatorname{GL}_r \to \operatorname{PGL}_r \to 1 $ to classify reductions, leading to isomorphisms $ E' \cong E \otimes L $ or $ E' \cong E^\vee \otimes L $.
• Leverage the fact that $ \operatorname{Aut}(M(r,\Lambda)) \to \operatorname{Aut}(\overline{M}(r,\Lambda)) $ is an isomorphism, since the smooth locus is $ M(r,\Lambda) $ and line bundles extend via cohomological isomorphisms.
• Use the triviality of the action on $ \operatorname{Pic}(M(r,\Lambda)) \cong \mathbb{Z} $ to show that automorphisms of $ M(r,\Lambda) $ extend to the compactified moduli space.

Experimental results

Research questions

• RQ1 What is the complete automorphism group of the moduli space $ M(r,\Lambda) $ of stable vector bundles on a curve of genus $ g \geq 4 $?
• RQ2 Can the automorphism group be described without relying on Hecke transforms or minimal rational curves?
• RQ3 How does the geometry of the Hitchin discriminant encode the automorphism structure of $ M(r,\Lambda) $?
• RQ4 Under what conditions does dualization of bundles appear as an automorphism of $ M(r,\Lambda) $?
• RQ5 To what extent does $ M(r,\Lambda) $ determine the underlying curve $ X $ and the rank $ r $?

Key findings

• The automorphism group of $ M(r,\Lambda) $ is generated by pullbacks via curve automorphisms $ \sigma $, tensorization with line bundles $ L $ satisfying $ L^r \otimes \sigma^*\Lambda \cong \Lambda $, and dualization when $ r \mid 2\deg(\Lambda) $.
• The proof avoids the use of minimal rational curves and Hecke transforms, relying instead on the Hitchin discriminant and nilpotent cone bundles.
• The automorphism group of the compactified moduli space $ \overline{M}(r,\Lambda) $ coincides with that of $ M(r,\Lambda) $, due to extension of line bundles and cohomological isomorphisms.
• For a generic stable bundle $ E $, any automorphism of $ M(r,\Lambda) $ induces an isomorphism $ E' \cong E \otimes L $ or $ E' \cong E^\vee \otimes L $, with $ L $ satisfying a degree condition.
• The Torelli-type theorem holds: if $ M_X(r,\Lambda) \cong M_{X'}(r',\Lambda') $, then $ X \cong X' $ and $ r = r' $, showing the moduli space classifies the curve and rank.
• The result generalizes to other moduli spaces, such as those of symplectic bundles, indicating broader applicability of the method.