Torelli theorem for moduli stacks of vector bundles and principal G-bundles

D. Alfaya, I. Biswas, T.L. Gómez, S. Mukhopadhyay

Journal of Geometry and Physics Vol. 207, pp. 105350-1 - 105350-15

Summary:

Given any irreducible smooth complex projective curve X, of genus at least 2, consider the moduli stack of vector bundles on X of fixed rank and determinant. It is proved that the isomorphism class of the stack uniquely determines the isomorphism class of the curve X and the rank of the vector bundles. The case of trivial determinant, rank 2 and genus 2 is specially interesting: the curve can be recovered from the moduli stack, but not from the moduli space (since this moduli space is P3 thus independently of the curve).

We also prove a Torelli theorem for moduli stacks of principal G-bundles on a curve of genus at least 3, where G is any non-abelian reductive group.

Spanish layman's summary:

Demostramos que la clase de isomorfismo de una curva compleja proyectiva esta determinada unívocamente por la clase de isomorfismo de su moduli stack de fibrados vectoriales. Se prueba también un Teorema de Torelli análogo para el moduli stack de G-fibrados principales.

English layman's summary:

We show that the isomorphism class of a complex projective curve is determined uniquely by the isomorphism class of its moduli stack of vector bundles. A similar Torelli Theorem is also proven for the moduli stack of principal G-bundles.

Keywords: Torelli theorem; Moduli stack; Higgs bundle; Hitchin map

JCR Impact Factor and WoS quartile: 1,600 - Q1 (2023)

DOI reference: https://doi.org/10.1016/j.geomphys.2024.105350

Published on paper: January 2025.

Published on-line: October 2024.

Citation:
D. Alfaya, I. Biswas, T.L. Gómez, S. Mukhopadhyay, Torelli theorem for moduli stacks of vector bundles and principal G-bundles. Journal of Geometry and Physics. Vol. 207, pp. 105350-1 - 105350-15, January 2025. [Online: October 2024]