Perspectives on the Construction and Compactification of Moduli Spaces

Laza, Radu

doi:10.1007/978-3-0348-0921-4_1

Cited by 9 publications

(7 citation statements)

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“…Gieseker [38] provided an alternative GIT construction of M g and its Deligne-Mumford compactification M g via stable curves as a quotient of the PGL N +1 -action on a suitable Hilbert scheme with a linearisation given by embedding in a Grassmannian associated to a sufficiently large choice of m. Although there is a direct proof that smooth curves are (asymptotically) GIT stable, the proof that stable curves are (asymptotically) GIT stable is indirect (see [72, §3.1]). Gieseker's Hilbert scheme construction is now the prevalent perspective, which has been generalised to give GIT constructions of moduli spaces of pointed stable curves and stable maps and their (birational) geometry is studied using VGIT (see [67,72]).…”

Section: 2mentioning

confidence: 99%

Moduli spaces and geometric invariant theory: old and new perspectives

Hoskins¹

2023

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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. This is even more tangible following the recent existence criteria of Alper-Halpern-Leistner-Heinloth [7], which equates the existence of moduli spaces to two simple valuative criteria and has been applied to various moduli problems [4,5,12,13].However, adequate and good moduli spaces are locally modelled on reductive GIT and require closed points to have reductive stabiliser groups. Thus, in some senses these two recent developments are orthogonal to each other and ideally there should eventually be an extension of non-reductive GIT to stacks.Acknowledgements. I am indebted to both Peter Newstead and Frances Kirwan, as I learned the basics of reductive GIT from Peter Newstead's Tata lecture notes [79] and discussions with Frances Kirwan. I am very grateful to Greg Bérczi, Dominic Bunnett, Eloise Hamilton, Josh Jackson and Frances Kirwan for numerous conversations on non-reductive GIT. I would also like to thank the organisers of VBAC 2022 for soliciting this paper in honour of Peter Newstead.Conventions. Throughout we will assume k is an algebraically closed field and all schemes are assumed to be finite type k-schemes, unless otherwise stated. By a point, we will mean a k-point (or equivalently, a closed point).

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Section: 2mentioning

confidence: 99%

Moduli spaces and geometric invariant theory: old and new perspectives

Hoskins¹

2023

Preprint

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“…Two significant exceptions are the period domains arising when considering moduli spaces of principally polarized abelian varieties and K3 surfaces. The Hermitian symmetric structure of D in these two cases, along with global Torelli theorems, is the underlying structure that has made Hodge theory such a powerful tool in the study of these moduli spaces and their compactifications (Laza, 2016). Even when the period domain D is not Hermitian, it may contain Hermitian symmetric Mumford-Tate subdomains D. (For example, every horizontal subdomain is Hermitian symmetric.)…”

Section: Motivationsmentioning

confidence: 99%

Hodge Representations

Han

Robles

2020

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Green–Griffiths–Kerr introduced Hodge representations to classify the Hodge groups of polarized Hodge structures, and the corresponding Mumford–Tate subdomains. We summarize how, given a fixed period domain $ \mathcal{D} $ , to enumerate the Hodge representations and corresponding Mumford–Tate subdomains $ D \subset \mathcal{D} $ . The procedure is illustrated in two examples: (i) weight two Hodge structures with $ {p}_g={h}^{2,0}=2 $ ; and (ii) weight three CY-type Hodge structures.

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“…Two significant exceptions are the period domains arising when considering moduli spaces of principally polarized abelian varieties and K3 surfaces. The Hermitian symmetric structure of D in these two cases, along with global Torelli theorems, is the underlying structure that has made Hodge theory such a powerful tool in the study of these moduli spaces and their compactifications [Laz16]. Even when the period domain D is not Hermitian, it may contain Hermitian symmetric Mumford-Tate subdomains D. (For example, every horizontal subdomain is Hermitian symmetric.)…”

Section: Motivations the Geometric Considerations Motivating A Classi...mentioning

confidence: 99%

Hodge Representations

Han,

Robles

2020

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Hodge representations were introduced by Green-Griffiths-Kerr to classify the Hodge groups of polarized Hodge structures, and the corresponding Mumford-Tate subdomains of a period domain. The purpose of this article is to provide an exposition of how, given a fixed period domain D, to enumerate the Hodge representations corresponding to Mumford-Tate subdomains D ⊂ D. After reviewing the well-known classical cases that D is Hermitian symmetric (weight n = 1, and weight n = 2 with p g = h 2,0 = 1), we illustrate this in the case that D is the period domain parameterizing polarized Hodge structures of (effective) weight two Hodge structures with first Hodge number p g = h 2,0 = 2. We also classify the Hodge representations of Calabi-Yau type, and enumerate the horizontal representations of CY 3-fold type. (The "horizontal" representations those with the property that corresponding domain D ⊂ D satisfies the infinitesimal period relation, a.k.a. Griffiths' transversality, and is therefore Hermitian.

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Perspectives on the Construction and Compactification of Moduli Spaces

Abstract: 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.

Cited by 9 publications

References 103 publications

Moduli spaces and geometric invariant theory: old and new perspectives

Moduli spaces and geometric invariant theory: old and new perspectives

Hodge Representations

Hodge Representations

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