scholarly journals Moduli spaces and locally symmetric varieties

2019 ◽  
Author(s):  
Eduard Looijenga
Keyword(s):  
Moduli Spaces ◽  
Symmetric Varieties

Birational geometry of the moduli space of quartic  surfaces

2019 ◽  
Vol 155 (9) ◽  
pp. 1655-1710
Author(s):  
Radu Laza ◽  
Kieran O’Grady
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Invariant Theory ◽  
Geometric Invariant Theory ◽  
Dimensional Case ◽  
Geometric Invariant ◽  
Type Iv ◽  
Log Canonical ◽  
Symmetric Varieties ◽  
The Relationship

By work of Looijenga and others, one understands the relationship between Geometric Invariant Theory (GIT) and Baily–Borel compactifications for the moduli spaces of degree-$2$ $K3$ surfaces, cubic fourfolds, and a few other related examples. The similar-looking cases of degree-$4$ $K3$ surfaces and double Eisenbud–Popescu–Walter (EPW) sextics turn out to be much more complicated for arithmetic reasons. In this paper, we refine work of Looijenga in order to handle these cases. Specifically, in analogy with the so-called Hassett–Keel program for the moduli space of curves, we study the variation of log canonical models for locally symmetric varieties of Type IV associated to $D$-lattices. In particular, for the $19$-dimensional case, we conjecturally obtain a continuous one-parameter interpolation between the GIT and Baily–Borel compactifications for the moduli of degree-$4$ $K3$ surfaces. The analogous $18$-dimensional case, which corresponds to hyperelliptic degree-$4$ $K3$ surfaces, can be verified by means of Variation of Geometric Invariant Theory (VGIT) quotients.

scholarly journals Moduli spaces of symmetric cubic fourfolds and locally symmetric varieties

2020 ◽  
Vol 14 (10) ◽  
pp. 2647-2683
Author(s):  
Chenglong Yu ◽  
Zhiwei Zheng
Keyword(s):  
Moduli Spaces ◽  
Symmetric Varieties

scholarly journals Moduli spaces of irreducible symplectic manifolds

2010 ◽  
Vol 146 (2) ◽  
pp. 404-434 ◽  
Author(s):  
V. Gritsenko ◽  
K. Hulek ◽  
G. K. Sankaran
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
General Type ◽  
Symplectic Manifolds ◽  
Symmetric Varieties

AbstractWe study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of polarised deformation K3[2]manifolds with polarisation of degree 2dand split type is of general type ifd≥12.

Smooth Compactifications of Locally Symmetric Varieties

2009 ◽  
Author(s):  
Avner Ash ◽  
David Mumford ◽  
Michael Rapoport ◽  
Yung-sheng Tai
Keyword(s):  
Symmetric Varieties

Geometry and Physics: Volume II

2018 ◽  
Keyword(s):  
Moduli Spaces ◽  
Mirror Symmetry ◽  
Geometric Analysis ◽  
Mathematical Physics ◽  
Poisson Geometry ◽  
Special Holonomy ◽  
Low Dimensional Topology ◽  
Generalized Complex Structures ◽  
Wide Range ◽  
Low Dimensional

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and Savilian Professor of Geometry at Oxford. The proceedings contain twenty-nine articles, including three by Fields medallists (Donaldson, Mori and Yau). The articles cover a wide range of topics in geometry and mathematical physics, including the following: Riemannian geometry, geometric analysis, special holonomy, integrable systems, dynamical systems, generalized complex structures, symplectic and Poisson geometry, low-dimensional topology, algebraic geometry, moduli spaces, Higgs bundles, geometric Langlands programme, mirror symmetry and string theory. These volumes will be of interest to researchers and graduate students both in geometry and mathematical physics.

scholarly journals On the Voevodsky motive of the moduli space of Higgs bundles on a curve

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Characteristic Zero ◽  
Rational Point ◽  
Triangulated Category ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
De Rham

AbstractWe study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

scholarly journals EXTREMAL CASES OF RAPOPORT–ZINK SPACES

2021 ◽  
pp. 1-56
Author(s):  
Ulrich Görtz ◽  
Xuhua He ◽  
Michael Rapoport
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Complete List ◽  
Qualitative Properties ◽  
Model Case ◽  
Divisible Groups

Abstract We investigate qualitative properties of the underlying scheme of Rapoport–Zink formal moduli spaces of p-divisible groups (resp., shtukas). We single out those cases where the dimension of this underlying scheme is zero (resp., those where the dimension is the maximal possible). The model case for the first alternative is the Lubin–Tate moduli space, and the model case for the second alternative is the Drinfeld moduli space. We exhibit a complete list in both cases.

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