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.

A boundary divisor in the moduli spaces of stable quintic surfaces

2017 ◽  
Vol 28 (04) ◽  
pp. 1750021 ◽  
Author(s):  
Julie Rana
Keyword(s):  
Algebraic Geometry ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Deformation Theory ◽  
General Type ◽  
Topological Invariants ◽  
Surfaces Of General Type ◽  
Wide Range ◽  
Stable Surfaces ◽  
Standing Question

We give a bound on which singularities may appear on Kollár–Shepherd-Barron–Alexeev stable surfaces for a wide range of topological invariants and use this result to describe all stable numerical quintic surfaces (KSBA-stable surfaces with [Formula: see text]) whose unique non-Du Val singularity is a Wahl singularity. We then extend the deformation theory of Horikawa to the log setting in order to describe the boundary divisor of the moduli space [Formula: see text] corresponding to these surfaces. Quintic surfaces are the simplest examples of surfaces of general type and the question of describing their moduli is a long-standing question in algebraic geometry.

scholarly journals Non-normal abelian covers

2012 ◽  
Vol 148 (4) ◽  
pp. 1051-1084 ◽  
Author(s):  
Valery Alexeev ◽  
Rita Pardini
Keyword(s):  
Abelian Group ◽  
Moduli Space ◽  
Moduli Spaces ◽  
General Type ◽  
Finite Abelian Group ◽  
Algebraic Varieties ◽  
Surfaces Of General Type ◽  
Normal Case ◽  
Abelian Cover ◽  
Abelian Covers

AbstractAn abelian cover is a finite morphism X→Y of varieties which is the quotient map for a generically faithful action of a finite abelian group G. Abelian covers with Y smooth and X normal were studied in [R. Pardini, Abelian covers of algebraic varieties, J. Reine Angew. Math. 417 (1991), 191–213; MR 1103912(92g:14012)]. Here we study the non-normal case, assuming that X and Y are S2 varieties that have at worst normal crossings outside a subset of codimension greater than or equal to two. Special attention is paid to the case of ℤr2-covers of surfaces, which is used in [V. Alexeev and R. Pardini, Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces, Preprint (2009), math.AG/arXiv:0901.4431] to construct explicitly compactifications of some components of the moduli space of surfaces of general type.

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 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.

scholarly journals THE HYPERELLIPTIC THETA MAP AND OSCULATING PROJECTIONS

2020 ◽  
pp. 1-23
Author(s):  
MICHELE BOLOGNESI ◽  
NÉSTOR FERNÁNDEZ VARGAS
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Hyperelliptic Curve ◽  
Geometric Description ◽  
Birational Model ◽  
Rank 2 ◽  
Semistable Vector Bundles ◽  
Ramification Locus

Abstract Let C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.

scholarly journals A note on moduli spaces of conformal classes for flat tori of higher dimension and on their conformal multiplication

2021 ◽  
Author(s):  
Anna Gori ◽  
Alberto Verjovsky ◽  
Fabio Vlacci
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Abelian Varieties ◽  
Complex Multiplication ◽  
Higher Dimension ◽  
Geometric Constructions ◽  
Flat Torus ◽  
Flat Tori

AbstractMotivated by the theory of complex multiplication of abelian varieties, in this paper we study the conformality classes of flat tori in $${\mathbb {R}}^{n}$$ R n and investigate criteria to determine whether a n-dimensional flat torus has non trivial (i.e. bigger than $${\mathbb {Z}}^{*}={\mathbb {Z}}{\setminus }\{0\}$$ Z ∗ = Z \ { 0 } ) semigroup of conformal endomorphisms (the analogs of isogenies for abelian varieties). We then exhibit several geometric constructions of tori with this property and study the class of conformally equivalent lattices in order to describe the moduli space of the corresponding tori.

scholarly journals NON-SYMPLECTIC INVOLUTIONS ON MANIFOLDS OF -TYPE

2020 ◽  
pp. 1-25
Author(s):  
CHIARA CAMERE ◽  
ALBERTO CATTANEO ◽  
ANDREA CATTANEO
Keyword(s):  
Moduli Spaces ◽  
Quadratic Forms ◽  
Symplectic Manifolds ◽  
Hilbert Schemes ◽  
Small Rank ◽  
Twisted Sheaves ◽  
Formula Type

We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant quadratic forms of the invariant and coinvariant lattice for the action of the involution on cohomology and explicitly describe the lattices in the cases where the invariant lattice has small rank. We also give a modular description of all $d$ -dimensional families of manifolds of $K3^{[n]}$ -type with a non-symplectic involution for $d\geqslant 19$ and $n\leqslant 5$ and provide examples arising as moduli spaces of twisted sheaves on a $K3$ surface.

scholarly journals On the boundary of the moduli spaces of log Hodge structures: Triviality of the torsor

2010 ◽  
Vol 198 ◽  
pp. 173-190 ◽  
Author(s):  
Tatsuki Hayama
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Hodge Structures

AbstractThis paper examines the moduli spaces of log Hodge structures introduced by Kato and Usui. This moduli space is a partial compactification of a discrete quotient of a period domain. This paper treats the following two cases: (A) where the period domain is Hermitian symmetric, and (B) where the Hodge structures are of the mirror quintic type. Especially it addresses a property of the torsor.

scholarly journals NEW EXAMPLES OF CALABI–YAU 3-FOLDS AND GENUS ZERO SURFACES

2014 ◽  
Vol 16 (02) ◽  
pp. 1350010 ◽  
Author(s):  
GILBERTO BINI ◽  
FILIPPO F. FAVALE ◽  
JORGE NEVES ◽  
ROBERTO PIGNATELLI
Keyword(s):  
Fundamental Group ◽  
Moduli Space ◽  
General Type ◽  
Picard Number ◽  
Surfaces Of General Type ◽  
Geometric Genus ◽  
Genus Zero ◽  
New Family ◽  
Hodge Numbers ◽  
Projective Lines

We classify the subgroups of the automorphism group of the product of four projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi–Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is nontrivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K2 = 3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.

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