A guide to moduli theory beyond GIT

Tomás L. Gómez; Andres Fernández Herrero; Alfonso Zamora

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arXiv · 2302.01871

A guide to moduli theory beyond GIT

Tomás L. Gómez·Andres Fernández Herrero·Alfonso Zamora

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Abstract

In this survey we provide an overview of some recent developments in the construction of moduli spaces using stack-theoretic techniques. We will also explain the analogue of Harder-Narasimhan stratifications for general stacks, known as $\Theta$-stratifications. As an application of the ideas exposed here, we address the moduli problem of principal bundles over higher dimensional projective varieties, as well as its different compactifications by the so-called principal $\rho$-sheaves. We construct a stratification by instability types whose lower strata admits a proper good moduli space of ``Gieseker semistable" objects and a new Gieseker-type Harder-Narasimhan filtration for these objects. Detailed proofs of the latter results will appear elsewhere.

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Tomás L. Gómez, Andres Fernández Herrero, Alfonso Zamora. 2023-02-03. A guide to moduli theory beyond GIT. https://arxiv.org/abs/2302.01871

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