Tropical Geometric Compactification of Moduli, II: A g Case and Holomorphic Limits

2018 ◽  
Vol 2019 (21) ◽  
pp. 6614-6660 ◽  
Author(s):  
Yuji Odaka
Keyword(s):  
Algebraic Curves ◽  
Abelian Varieties ◽  
Moduli Of Curves ◽  
Complex Dimension ◽  
Crucial Difference ◽  
Flat Tori ◽  
Hausdorff Limits

Abstract We compactify the classical moduli variety Ag of principally polarized abelian varieties of complex dimension g, by attaching the moduli of flat tori of real dimensions at most g in an explicit manner. Equivalently, we explicitly determine the Gromov–Hausdorff limits of principally polarized abelian varieties. This work is analogous to [50], where we compactified the moduli of curves by attaching the moduli of metrized graphs. Then, we also explicitly specify the Gromov–Hausdorff limits along holomorphic families of abelian varieties and show that these form special nontrivial subsets of the whole boundary. We also do the same for algebraic curves case and observe a crucial difference with the case of abelian varieties.

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 Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: An algebro-geometric approach

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Enrico Arbarello ◽  
Giulio Codogni ◽  
Giuseppe Pareschi
Keyword(s):  
Linear System ◽  
Line Bundle ◽  
Abelian Variety ◽  
Algebraic Curves ◽  
Abelian Varieties ◽  
Geometric Approach ◽  
Kp Equation ◽  
Basic Tool ◽  
Base Locus

Abstract We give completely algebro-geometric proofs of a theorem by T. Shiota, and of a theorem by I. Krichever, characterizing Jacobians of algebraic curves among all irreducible principally polarized abelian varieties. Shiota’s characterization is given in terms of the KP equation. Krichever’s characterization is given in terms of trisecant lines to the Kummer variety. Here we treat the case of flexes and degenerate trisecants. The basic tool we use is a theorem we prove asserting that the base locus of the linear system associated to an effective line bundle on an abelian variety is reduced. This result allows us to remove all the extra assumptions that were introduced in the theorems by the first author, C. De Concini, G.Marini, and O. Debarre, in order to achieve algebro-geometric proofs of the results above.

Hypergeometric Abelian Varieties

2003 ◽  
Vol 55 (5) ◽  
pp. 897-932 ◽  
Author(s):  
Natália Archinard
Keyword(s):  
Hypergeometric Series ◽  
Algebraic Curves ◽  
Abelian Varieties ◽  
Geometric Construction ◽  
Exceptional Sets ◽  
Monodromy Groups

AbstractIn this paper, we construct abelian varieties associated to Gauss’ and Appell–Lauricella hypergeometric series. Abelian varieties of this kind and the algebraic curves we define to construct them were considered by several authors in settings ranging from monodromy groups (Deligne, Mostow), exceptional sets (Cohen, Wolfart, Wüstholz), modular embeddings (Cohen, Wolfart) to CM-type (Cohen, Shiga, Wolfart) and modularity (Darmon). Our contribution is to provide a complete, explicit and self-contained geometric construction.

scholarly journals Manifolds with Many Rarita–Schwinger Fields

2021 ◽  
Author(s):  
Christian Bär ◽  
Rafe Mazzeo
Keyword(s):  
Dirac Operator ◽  
Compact Manifolds ◽  
Overdetermined Problem ◽  
Complex Dimension ◽  
Linearly Independent ◽  
Twisted Dirac Operator ◽  
Flat Tori ◽  
Divergence Free ◽  
Simply Connected ◽  
Einstein Constant

AbstractThe Rarita–Schwinger operator is the twisted Dirac operator restricted to $$\nicefrac 32$$ 3 2 -spinors. Rarita–Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions. In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita–Schwinger fields tends to infinity. These manifolds are either simply connected Kähler–Einstein spin with negative Einstein constant, or products of such spaces with flat tori. Moreover, we construct Calabi–Yau manifolds of even complex dimension with more linearly independent Rarita–Schwinger fields than flat tori of the same dimension.

Moduli spaces for real algebraic curves and real abelian varieties

1989 ◽  
Vol 201 (2) ◽  
pp. 151-165 ◽  
Author(s):  
M. Sepp�l� ◽  
R. Silhol
Keyword(s):  
Moduli Spaces ◽  
Algebraic Curves ◽  
Abelian Varieties ◽  
Real Algebraic Curves
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