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Doctoral thesis
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Wall-crossings for moduli spaces of parabolic bundles on curves

ContributorsTrapeznikova, Olgaorcid
Imprimatur date2023
Defense date2023
Abstract

The Verlinde formula, an expression for the Euler characteristic of line bundles on the moduli spaces of stable bundles on a curve, is a strikingly beautiful statement in enumerative geometry motivated by quantum physics. It has attracted a lot of attention over the years, and has a number of different proofs.

In Chapter 2 of this thesis, we give a new proof of the more difficult parabolic variant of this formula based on a comparison of wall-crossings in Geometric Invariant Theory and certain iterated residues calculus. On the way, we develop a tautological variant of Hecke correspondences, calculate the Hilbert polynomials of the moduli spaces, and present a new, transparent approach to the rho-shift problem of the theory.

In Chapter 3 we show that the residue/wall-crossing methods of Chapter 2 may be successfully employed to describe the pushforward maps in the K-theory of moduli spaces and present new, explicit formulas for the Euler characteristic of a wider class of vector bundles on the moduli space of stable parabolic bundles.

Our work was motivated by the results of Teleman and Woodward on the index of K- theory classes of moduli stacks.

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Citation (ISO format)
TRAPEZNIKOVA, Olga. Wall-crossings for moduli spaces of parabolic bundles on curves. 2023. doi: 10.13097/archive-ouverte/unige:172123
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