scholarly journals Semicontinuity of Gauss maps and the Schottky problem

2021 ◽  
Author(s):  
Giulio Codogni ◽  
Thomas Krämer
Keyword(s):  
Abelian Variety ◽  
Moduli Space ◽  
Cotangent Bundle ◽  
Abelian Varieties ◽  
Topological Type ◽  
Intersection Cohomology ◽  
Theta Divisor ◽  
Schottky Problem ◽  
Gauss Maps

AbstractWe show that the degree of Gauss maps on abelian varieties is semicontinuous in families, and we study its jump loci. As an application we obtain that in the case of theta divisors this degree answers the Schottky problem. Our proof computes the degree of Gauss maps by specialization of Lagrangian cycles on the cotangent bundle. We also get similar results for the intersection cohomology of varieties with a finite morphism to an abelian variety; it follows that many components of Andreotti–Mayer loci, including the Schottky locus, are part of the stratification of the moduli space of ppav’s defined by the topological type of the theta divisor.

scholarly journals Characteristic cycles and the microlocal geometry of the Gauss map, II

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Thomas Krämer
Keyword(s):  
Weak Solution ◽  
Weyl Group ◽  
Lie Algebras ◽  
Abelian Variety ◽  
Cotangent Bundle ◽  
Abelian Varieties ◽  
Gauss Map ◽  
Schottky Problem ◽  
Group Orbit

Abstract We show that any Weyl group orbit of weights for the Tannakian group of semisimple holonomic 𝒟 {{\mathscr{D}}} -modules on an abelian variety is realized by a Lagrangian cycle on the cotangent bundle. As applications we discuss a weak solution to the Schottky problem in genus five, an obstruction for the existence of summands of subvarieties on abelian varieties, and a criterion for the simplicity of the arising Lie algebras.

Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves

2014 ◽  
Vol 66 (5) ◽  
pp. 1167-1200 ◽  
Author(s):  
Victor Rotger ◽  
Carlos de Vera-Piquero
Keyword(s):  
Abelian Variety ◽  
Moduli Space ◽  
Quadratic Field ◽  
Galois Representations ◽  
Abelian Varieties ◽  
Main Idea ◽  
Galois Representation ◽  
Rational Points ◽  
Shimura Curves ◽  
Fields Of Moduli

AbstractThe purpose of this note is to introduce a method for proving the non-existence of rational points on a coarse moduli space X of abelian varieties over a given number field K in cases where the moduli problem is not fine and points in X(K) may not be represented by an abelian variety (with additional structure) admitting a model over the field K. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired by the work of Ellenberg and Skinner on the modularity of ℚ-curves, is that one may still attach a Galois representation of Gal(/K) with values in the quotient group GL(Tℓ(A))/ Aut(A) to a point P = [A] ∈ X(K) represented by an abelian variety A/, provided Aut(A) lies in the centre of GL(Tℓ(A)). We exemplify our method in the cases where X is a Shimura curve over an imaginary quadratic field or an Atkin–Lehner quotient over ℚ.

scholarly journals On the dimension of voisin sets in the moduli space of abelian varieties

2021 ◽  
Author(s):  
E. Colombo ◽  
J. C. Naranjo ◽  
G. P. Pirola
Keyword(s):  
Abelian Variety ◽  
Moduli Space ◽  
Abelian Varieties ◽  
Chow Group ◽  
Hyperelliptic Locus

AbstractWe study the subsets $$V_k(A)$$ V k ( A ) of a complex abelian variety A consisting in the collection of points $$x\in A$$ x ∈ A such that the zero-cycle $$\{x\}-\{0_A\}$$ { x } - { 0 A } is k-nilpotent with respect to the Pontryagin product in the Chow group. These sets were introduced recently by Voisin and she showed that $$\dim V_k(A) \le k-1$$ dim V k ( A ) ≤ k - 1 and $$\dim V_k(A)$$ dim V k ( A ) is countable for a very general abelian variety of dimension at least $$2k-1$$ 2 k - 1 . We study in particular the locus $${\mathcal {V}}_{g,2}$$ V g , 2 in the moduli space of abelian varieties of dimension g with a fixed polarization, where $$V_2(A)$$ V 2 ( A ) is positive dimensional. We prove that an irreducible subvariety $${\mathcal {Y}} \subset {\mathcal {V}}_{g,2}$$ Y ⊂ V g , 2 , $$g\ge 3$$ g ≥ 3 , such that for a very general $$y \in {\mathcal {Y}}$$ y ∈ Y there is a curve in $$V_2(A_y)$$ V 2 ( A y ) generating A satisfies $$\dim {\mathcal {Y}}\le 2g - 1.$$ dim Y ≤ 2 g - 1 . The hyperelliptic locus shows that this bound is sharp.

scholarly journals LINE BUNDLES OF TYPE (1,…,1,2,…,2,4,…,4) ON ABELIAN VARIETIES

2001 ◽  
Vol 12 (01) ◽  
pp. 125-142
Author(s):  
JAYA N. IYER
Keyword(s):  
Linear System ◽  
Abelian Variety ◽  
Moduli Space ◽  
Abelian Varieties ◽  
Line Bundles ◽  
Complete Linear System

We show birationality of the morphism associated to line bundles L of type (1,…,1,2,…,2,4,…,4) on a generic g-dimensional abelian variety into its complete linear system such that h0(L) = 2g. When g = 3, we describe the image of the abelian threefold and from the geometry of the moduli space SUC(2) in the linear system |2θC|, we obtain analogous results in ℙH0(L).

ON MODULI SPACE OF HIGGS Gp(2n, ℂ)-BUNDLES OVER A RIEMANN SURFACE

2010 ◽  
Vol 07 (02) ◽  
pp. 311-322
Author(s):  
INDRANIL BISWAS ◽  
SARBESWAR PAL
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Hilbert Scheme ◽  
Symplectic Form ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Topological Type ◽  
Total Space ◽  
The Other ◽  
Complex Variety

Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denoted by KX. Let [Formula: see text] denote the moduli space of semistable Higgs Gp (2n, ℂ)-bundles over X of fixed topological type. The complex variety [Formula: see text] has a natural holomorphic symplectic structure. On the other hand, for any ℓ ≥ 1, the Liouville symplectic from on the total space of KX defines a holomorphic symplectic structure on the Hilbert scheme Hilb ℓ(KX) parametrizing the zero-dimensional subschemes of KX. We relate the symplectic form on Hilb ℓ(KX) with the symplectic form on [Formula: see text].

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 Donaldson–Thomas invariants versus intersection cohomology of quiver moduli

2019 ◽  
Vol 2019 (754) ◽  
pp. 143-178 ◽  
Author(s):  
Sven Meinhardt ◽  
Markus Reineke
Keyword(s):  
Moduli Space ◽  
Stability Condition ◽  
Intersection Cohomology ◽  
Quiver Representations ◽  
Compactly Supported ◽  
Generic Stability ◽  
Relative Version ◽  
Zero Potential ◽  
Stable Locus

Abstract The main result of this paper is the statement that the Hodge theoretic Donaldson–Thomas invariant for a quiver with zero potential and a generic stability condition agrees with the compactly supported intersection cohomology of the closure of the stable locus inside the associated coarse moduli space of semistable quiver representations. In fact, we prove an even stronger result relating the Donaldson–Thomas “function” to the intersection complex. The proof of our main result relies on a relative version of the integrality conjecture in Donaldson–Thomas theory. This will be the topic of the second part of the paper, where the relative integrality conjecture will be proven in the motivic context.

ON THE RANK OF ABELIAN VARIETIES OVER AMPLE FIELDS

2010 ◽  
Vol 06 (03) ◽  
pp. 579-586 ◽  
Author(s):  
ARNO FEHM ◽  
SEBASTIAN PETERSEN
Keyword(s):  
Finite Field ◽  
Abelian Variety ◽  
Galois Extension ◽  
Characteristic Zero ◽  
Rational Point ◽  
Abelian Varieties ◽  
Finitely Generated ◽  
Infinite Rank ◽  
Finite Set

A field K is called ample if every smooth K-curve that has a K-rational point has infinitely many of them. We prove two theorems to support the following conjecture, which is inspired by classical infinite rank results: Every non-zero Abelian variety A over an ample field K which is not algebraic over a finite field has infinite rank. First, the ℤ(p)-module A(K) ⊗ ℤ(p) is not finitely generated, where p is the characteristic of K. In particular, the conjecture holds for fields of characteristic zero. Second, if K is an infinite finitely generated field and S is a finite set of local primes of K, then every Abelian variety over K acquires infinite rank over certain subfields of the maximal totally S-adic Galois extension of K. This strengthens a recent infinite rank result of Geyer and Jarden.

scholarly journals ON SELMER RANK PARITY OF TWISTS

2016 ◽  
Vol 102 (3) ◽  
pp. 316-330 ◽  
Author(s):  
MAJID HADIAN ◽  
MATTHEW WEIDNER
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Arbitrary Number ◽  
Hyperelliptic Curve ◽  
Abelian Varieties ◽  
Hyperelliptic Curves ◽  
Torsion Subgroup ◽  
Tate Conjecture ◽  
Quadratic Twists

In this paper we study the variation of the $p$-Selmer rank parities of $p$-twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$-rational $p$-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

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