Hilbert-Mumford stability on algebraic stacks and applications to $\mathcal{G}$-bundles on curves

Abstract

In these notes we reformulate the classical Hilbert-Mumford criterion for GIT stability in terms of algebraic stacks, this was independently done by Halpern-Leinster. We also give a geometric condition that guarantees the existence of separated coarse moduli spaces for the substack of stable objects. This is then applied to construct coarse moduli spaces for torsors under parahoric group schemes over curves. Comment: 37 pages

Very stable Higgs bundles, equivariant multiplicity and mirror symmetry

“…Proof The proof is by a Higgs bundle analogue of the Rees construction. The vector bundle version is discussed in [3,29,35]. First let us assume that we have a filtration (3.6) which is compatible with the Higgs field i.e.…”

Moduli spaces and geometric invariant theory: old and new perspectives

“…We also outline another significant development that extends ideas of (reductive) GIT to stacks, as pioneered by Alper, 48,7,52]. Alper's notion of good and adequate moduli spaces of stacks enables GIT-free constructions of moduli spaces.…”

“…Stability and existence criteria. Halpern-Leistner [46] and Heinloth [52] studied how ideas in reductive GIT, such as the Hilbert-Mumford criterion, can be applied to stacks. The role of 1-PSs and their limits can be replaced by the stack Theta:…”

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Many moduli spaces are constructed as quotients of group actions; this paper surveys the classical theory, as well as recent progress and applications. We review geometric invariant theory for reductive groups and how it is used to construct moduli spaces, and explain two new developments extending this theory to non-reductive groups and to stacks, which enable the construction of new moduli spaces.

VICTORIA HOSKINSWe also outline another significant development that extends ideas of (reductive) GIT to stacks, as pioneered by Alper, 48,7,52]. Alper's notion of good and adequate moduli spaces of stacks enables GIT-free constructions of moduli spaces.

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A general notion of coherent systems

“…On the other hand, we will explain in Section 4 how the approach 𝛿 → 1/(𝑎 • (𝑎 − 1) • 𝑟 𝑎−1 ) corresponds to the process of letting the stability parameter in [36] tend to infinity. To conclude this introduction, let us point out that new versions of geometric invariant theory have been developed which are based on the theory of stacks ( [21], [1]) or affine Graßmannians [18]. In situations where a traditional approach via geometric invariant theory can be carried out, these techniques will lead to the same basic conclusions.…”