scholarly journals Singular fibers of very general Lagrangian fibrations

2021 ◽  
pp. 2150070
Author(s):  
Justin Sawon
Keyword(s):  
Moduli Space ◽  
Abelian Varieties ◽  
Partial Result ◽  
Codimension One ◽  
Lagrangian Fibration ◽  
Singular Fibers

Let [Formula: see text] be a (holomorphic) Lagrangian fibration that is very general in the moduli space of Lagrangian fibrations. We conjecture that the singular fibers in codimension one must be semistable degenerations of abelian varieties. We prove a partial result towards this conjecture, and describe an example that provides further evidence.

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 Codimension One Holomorphic Distributions on the Projective Three-space

2018 ◽  
Vol 2020 (23) ◽  
pp. 9011-9074 ◽  
Author(s):  
Omegar Calvo-Andrade ◽  
Maurício Corrêa ◽  
Marcos Jardim
Keyword(s):  
Projective Space ◽  
Moduli Space ◽  
Tangent Bundle ◽  
Projective Variety ◽  
Arbitrary Degree ◽  
Codimension One ◽  
Quot Scheme ◽  
Space Of Distributions ◽  
Three Space

Abstract We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck’s Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. Finally, we prove that every rational foliation and certain logarithmic foliations have stable tangent sheaves.

scholarly journals Totally geodesic subvarieties in the moduli space of curves

2020 ◽  
pp. 2050020
Author(s):  
Alessandro Ghigi ◽  
Gian Pietro Pirola ◽  
Sara Torelli
Keyword(s):  
Moduli Space ◽  
Abelian Varieties ◽  
Moduli Space Of Curves ◽  
Totally Geodesic

In this paper, we study totally geodesic subvarieties [Formula: see text] of the moduli space of principally polarized abelian varieties with respect to the Siegel metric, for [Formula: see text]. We prove that if [Formula: see text] is generically contained in the Torelli locus, then [Formula: see text].

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