scholarly journals Extending the Prym map to toroidal compactifications of the moduli space of abelian varieties (with an appendix by Mathieu Dutour Sikirić)

2017 ◽  
Vol 19 (3) ◽  
pp. 659-723
Author(s):  
Sebastian Casalaina-Martin ◽  
Samuel Grushevsky ◽  
Klaus Hulek ◽  
Radu Laza
Keyword(s):  
Moduli Space ◽  
Abelian Varieties ◽  
Toroidal Compactifications ◽  
Prym Map

scholarly journals Generic Torelli theorem for Prym varieties of ramified coverings

2012 ◽  
Vol 148 (4) ◽  
pp. 1147-1170 ◽  
Author(s):  
Valeria Ornella Marcucci ◽  
Gian Pietro Pirola
Keyword(s):  
Moduli Space ◽  
Abelian Varieties ◽  
Branch Points ◽  
Prym Variety ◽  
Prym Varieties ◽  
Torelli Theorem ◽  
Image Position ◽  
Prym Map ◽  
Double Coverings ◽  
Ramified Coverings

AbstractWe consider the Prym map from the space of double coverings of a curve of genus gwithrbranch points to the moduli space of abelian varieties. We prove that 𝒫:ℛg,r→𝒜δg−1+r/2is generically injective ifWe also show that a very general Prym variety of dimension at least 4 is not isogenous to a Jacobian.

scholarly journals Stable cohomology of the perfect cone toroidal compactification of 𝒜 g

2018 ◽  
Vol 2018 (741) ◽  
pp. 211-254 ◽  
Author(s):  
Samuel Grushevsky ◽  
Klaus Hulek ◽  
Orsola Tommasi
Keyword(s):  
Moduli Space ◽  
Polynomial Algebra ◽  
Abelian Varieties ◽  
Fixed Degree ◽  
Toroidal Compactifications ◽  
Stable Cohomology

Abstract We show that the cohomology of the perfect cone (also called first Voronoi) toroidal compactification {{{\mathcal{A}}_{g}^{\operatorname{Perf}}}} of the moduli space of complex principally polarized abelian varieties stabilizes in close to the top degree. Moreover, we show that this stable cohomology is purely algebraic, and we compute it in degree up to 13. Our explicit computations and stabilization results apply in greater generality to various toroidal compactifications and partial compactifications, and in particular we show that the cohomology of the matroidal partial compactification {{{\mathcal{A}}_{g}^{\operatorname{Matr}}}} stabilizes in fixed degree, and forms a polynomial algebra. For degree up to 8, we describe explicitly the generators of the cohomology, and discuss various approaches to computing all of the stable cohomology in general.

scholarly journals Shimura subvarieties in the Prym locus of ramified Galois coverings

2021 ◽  
Author(s):  
Gian Paolo Grosselli ◽  
Abolfazl Mohajer
Keyword(s):  
Computer Algebra ◽  
Group Action ◽  
Moduli Space ◽  
Abelian Varieties ◽  
Fixed Group ◽  
Galois Coverings ◽  
Base Curve ◽  
Prym Map

AbstractWe study Shimura (special) subvarieties in the moduli space $$A_{p,D}$$ A p , D of complex abelian varieties of dimension p and polarization type D. These subvarieties arise from families of covers compatible with a fixed group action on the base curve such that the quotient of the base curve by the group is isomorphic to $${{\mathbb {P}}}^1$$ P 1 . We give a criterion for the image of these families under the Prym map to be a special subvariety and, using computer algebra, obtain 210 Shimura subvarieties contained in the Prym locus.

scholarly journals Stable Betti Numbers of (Partial) Toroidal Compactifications of the Moduli Space of Abelian Varieties

2018 ◽  
pp. 581-610 ◽  
Author(s):  
Samuel Grushevsky ◽  
Klaus Hulek ◽  
Orsola Tommasi ◽  
Mathieu Dutour Sikirić
Keyword(s):  
Moduli Space ◽  
Automorphism Groups ◽  
Abelian Varieties ◽  
Betti Numbers ◽  
Cohomology Algebra ◽  
Algebra Structure ◽  
Toroidal Compactifications ◽  
Number Of Generators ◽  
Stable Cohomology ◽  
Finiteness Properties

This chapter presents an algorithm for explicitly computing the number of generators of the stable cohomology algebra of any rationally smooth partial toroidal compactification of Ag, satisfying certain additivity and finiteness properties, in terms of the combinatorics of the corresponding toric fans. In particular, the algorithm determines the stable cohomology of the matroidal partial compactification, in terms of simple regular matroids that are irreducible with respect to the 1-sum operation, and their automorphism groups. The algorithm also applies to compute the stable Betti numbers in close to top degree for the perfect cone toroidal compactification. This suggests the existence of an algebra structure on the stable cohomology of the perfect cone compactification in close to top degree.

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.

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