- CiteKey: “Laz14” - Type: report - Title: “Perspectives on the construction and compactification of moduli spaces,” - Author: “Laza, Radu;” - Publisher: “arXiv,” - Year: 2014 - Keywords: “Mathematics - Algebraic Geometry”

Perspectives on the construction and compactification of moduli spaces

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• URL: http://arxiv.org/abs/1403.2105
• URI: http://zotero.org/users/1049732/items/29T44PL8
• Open in Zotero: Zotero

Abstract

In these notes, we introduce various approaches (GIT, Hodge theory, and KSBA) to constructing and compactifying moduli spaces. We then discuss the pros and cons for each approach, as well as some connections between them.

Extracted Annotations

Notes

• Model examples: \(\mkern 1.5mu\overline{\mkern-1.5mu{\mathcal{M}_g}\mkern-1.5mu}\mkern 1.5mu\) and \(\mkern 1.5mu\overline{\mkern-1.5mu{\mathcal{A}_g}\mkern-1.5mu}\mkern 1.5mu\)
• Standard approaches to compactifying: GIT, GHS, MMP, SBB.
• GIT:
  • Embed into some \({\mathbf{P}}^N\) and get \({\mathcal{M}}\) from a Hilbert scheme
  • Forget the embedding via a quotient \(X{ \mathbin{/\mkern-6mu/}}G\) for some \(G\), usually \(G = \operatorname{PGL}_{N+1}\).
  • Downside: hard to understand \((X{ \mathbin{/\mkern-6mu/}}G)^{{\mathrm{ss}}}\)
• VHS:
  • Put Hodge structures on \(H^*(X; R)\) for \(R={\mathbf{Z}}, {\mathbf{C}}\) and get a period domain \({\mathbb{D}}\) which is a moduli of Hodge structures.
  • Form a moduli stack, get a period map \({\mathcal{P}}: {\mathcal{M}}\to {\mathbb{D}}/\Gamma\) where quotienting by \(\Gamma\) forgets the marking.
  • Torelli theorems yield that \({\mathcal{P}}\) is injective, when is it bijective?
  • Griffiths transversality yields that the periods are constrained by systems of ODEs.
  • Main issues are circumvented for PPAVs and K3s.
  • Note: Looijenga compares GIT and VHS.
• MMP:
  • Glue pathes into a stack, choose degenerations to show it is proper and separated.
  • By the valuative criteria, it suffices to consider 1-parameter degenerations.
  • Issues with uniqueness of the central fiber, but canonical models for general type are unique.
  • Allowing singularities yields uniqueness of limits.
• SBB (Satake-Bailey-Borel):
  • For locally symmetric varieties \(D/\Gamma\), compactify to \(\mathop{\mathrm{Proj}}A(\Gamma)\) for \(\Gamma\) a certain ring of automorphism forms.
  • Get a toric compactification by forgetting a marking? \((D/\Gamma)^{\Sigma} \to \mathop{\mathrm{Proj}}A(\Gamma)\).