Theory of Degeneration for Polarized Abelian Schemes

2013 ◽  
Author(s):  
Kai-Wen Lan
Keyword(s):  
Abelian Varieties ◽  
Degenerate Case ◽  
Analytic Construction ◽  
Abelian Schemes ◽  
Special Case

This chapter reproduces the theory of degeneration data for polarized abelian varieties, following D. Mumford's 1972 monograph, An analytic construction of degenerating abelian varieties over complete rings, as well as the first three chapters of the Degeneration of abelian varieties (1990), by G. Faltings and C.-L. Chai. Although there is essentially nothing new in this chapter, some modifications have been introduced to make the statements compatible with a certain understanding of the proofs. Moreover, since Mumford and Faltings and Chai have supplied full details only in the completely degenerate case, this chapter balances the literature by avoiding the special case and treating all cases equally.

scholarly journals Trajectory analysis and extrapolation in barrier function methods

1978 ◽  
Vol 20 (3) ◽  
pp. 352-369 ◽  
Author(s):  
Krisorn Jittorntrum ◽  
M. R. Osborne
Keyword(s):  
Barrier Function ◽  
Trajectory Analysis ◽  
Degenerate Case ◽  
Extrapolation Methods ◽  
Extrapolation Procedure ◽  
Sensitivity Measure ◽  
Successive Solution ◽  
Degeneracy Condition ◽  
Special Case ◽  
Solution Estimates

AbstractIt has been known for some time that if a certain non-degeneracy condition is satisfied then the successive solution estimates x(r) produced by barrier function techniques lie on a smooth trajectory. Accordingly, extrapolation methods can be used to calculate x(0). In this paper we analyse the situation further treating the special case of the log barrier function. If the non-degeneracy assumption is not satisfied then the approach to x(0) is like r½ rather than like r which would be expected in the non-degenerate case. A measure of sensitivity is introduced which becomes large when the non- degeneracy assumption is close to violation, and it is shown that this sensitivity measure is related to the growth of dix/dri with respect to i for fixed r small enough on the solution trajectory. With this information it is possible to analyse the extrapolation procedure and to predict the number of stages of extrapolation which are useful.

Structures of Semi-Abelian Schemes

2013 ◽  
Author(s):  
Kai-Wen Lan
Keyword(s):  
Abelian Varieties ◽  
Fundamental Property ◽  
Fundamental Importance ◽  
Multiplicative Type ◽  
Abelian Schemes

This chapter explains well-known notions important for the study of semi-abelian schemes. It first studies groups of multiplicative type and the torsors under them. A fundamental property of groups of multiplicative type is that they are rigid in the sense that they cannot be deformed. The chapter then turns to biextensions, cubical structures, semi-abelian schemes, Raynaud extensions, and certain dual objects for the last two notions extending the notion of dual abelian varieties. Such notions are, as this chapter shows, of fundamental importance in the study of the degeneration of abelian varieties. The main objective here is to understand the statement and the proof of the theory of degeneration data.

scholarly journals On the Explicite Defining Relations of Abelian Schemes of Level Three

1966 ◽  
Vol 27 (1) ◽  
pp. 143-157 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Abelian Varieties ◽  
Dimensional Case ◽  
Plane Curves ◽  
Complex Numbers ◽  
Defining Relations ◽  
Higher Dimensional ◽  
Dimension One ◽  
Abelian Schemes

It is known classically that abelian varieties of dimension one over the field of complex numbers may be expressed by non-singular Hesse’s canonical cubic plane curves, The purpose of the present paper is to generalize this idea to higher dimensional case.

scholarly journals Skew Killing spinors in four dimensions

2021 ◽  
Author(s):  
Nicolas Ginoux ◽  
Georges Habib ◽  
Ines Kath
Keyword(s):  
Degenerate Case ◽  
Killing Spinor ◽  
Warped Products ◽  
Riemannian Product ◽  
Four Dimensions ◽  
Killing Spinors ◽  
Local Isometry ◽  
Spin Manifolds ◽  
Special Case

AbstractThis paper is devoted to the classification of 4-dimensional Riemannian spin manifolds carrying skew Killing spinors. A skew Killing spinor $$\psi $$ ψ is a spinor that satisfies the equation $$\nabla _X\psi =AX\cdot \psi $$ ∇ X ψ = A X · ψ with a skew-symmetric endomorphism A. We consider the degenerate case, where the rank of A is at most two everywhere and the non-degenerate case, where the rank of A is four everywhere. We prove that in the degenerate case the manifold is locally isometric to the Riemannian product $${\mathbb {R}}\times N$$ R × N with N having a skew Killing spinor and we explain under which conditions on the spinor the special case of a local isometry to $${\mathbb {S}}^2\times {\mathbb {R}}^2$$ S 2 × R 2 occurs. In the non-degenerate case, the existence of skew Killing spinors is related to doubly warped products whose defining data we will describe.

scholarly journals Trace of abelian varieties over function fields and the geometric Bogomolov conjecture

2018 ◽  
Vol 2018 (741) ◽  
pp. 133-159
Author(s):  
Kazuhiko Yamaki
Keyword(s):  
Abelian Variety ◽  
Function Field ◽  
Function Fields ◽  
Abelian Varieties ◽  
Constant Field ◽  
Abelian Schemes ◽  
Bogomolov Conjecture

Abstract We prove that the geometric Bogomolov conjecture for any abelian varieties is reduced to that for nowhere degenerate abelian varieties with trivial trace. In particular, the geometric Bogomolov conjecture holds for abelian varieties whose maximal nowhere degenerate abelian subvariety is isogenous to a constant abelian variety. To prove the results, we investigate closed subvarieties of abelian schemes over constant varieties, where constant varieties are varieties over a function field which can be defined over the constant field of the function field.

scholarly journals Rank 2 local systems and abelian varieties

2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Raju Krishnamoorthy ◽  
Ambrus Pál
Keyword(s):  
Abelian Varieties ◽  
Strong Form ◽  
Projective Varieties ◽  
Local Systems ◽  
Projective Curves ◽  
Rank 2 ◽  
Divisible Groups ◽  
Abelian Schemes ◽  
Formula Type

AbstractLet $$X/\mathbb {F}_{q}$$ X / F q be a smooth, geometrically connected variety. For X projective, we prove a Lefschetz-style theorem for abelian schemes of $$\text {GL}_2$$ GL 2 -type on X, modeled after a theorem of Simpson. Inspired by work of Corlette-Simpson over $$\mathbb {C}$$ C , we formulate a conjecture that absolutely irreducible rank 2 local systems with infinite monodromy on X come from families of abelian varieties. We have the following application of our main result. If one assumes a strong form of Deligne’s (p-adic) companions conjecture from Weil II, then our conjecture for projective varieties reduces to the conjecture for projective curves. We also answer affirmitavely a question of Grothendieck on extending abelian schemes via their p-divisible groups.

scholarly journals Abelian surfaces good away from 2

2017 ◽  
Vol 13 (04) ◽  
pp. 991-1001
Author(s):  
Christopher Rasmussen ◽  
Akio Tamagawa
Keyword(s):  
Elliptic Curves ◽  
Number Field ◽  
High Dimension ◽  
Abelian Varieties ◽  
Number Fields ◽  
Sufficient Condition ◽  
Good Reduction ◽  
Abelian Surfaces ◽  
Special Case

Fix a number field [Formula: see text] and a rational prime [Formula: see text]. We consider abelian varieties whose [Formula: see text]-power torsion generates a pro-[Formula: see text] extension of [Formula: see text] which is unramified away from [Formula: see text]. It is a necessary, but not generally sufficient, condition that such varieties have good reduction away from [Formula: see text]. In the special case of [Formula: see text], we demonstrate that for abelian surfaces [Formula: see text], good reduction away from [Formula: see text] does suffice. The result is extended to elliptic curves and abelian surfaces over certain number fields unramified away from [Formula: see text]. An explicit example is constructed to demonstrate that good reduction away from [Formula: see text] is not sufficient, at [Formula: see text], for abelian varieties of sufficiently high dimension.

Unlikely Intersections in Elliptic Surfaces and Problems of Masser

2012 ◽  
Author(s):  
Umberto Zannier ◽  
David Masser
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Multiplicative Group ◽  
Abelian Varieties ◽  
Elliptic Surfaces ◽  
Group Context ◽  
Special Case ◽  
Relative Case

This chapter turns from the multiplicative-group context to the context of abelian varieties. There are here entirely similar results and conjectures: we have already recalled the Manin–Mumford conjecture, and pointed out that the Zilber conjecture also admits an abelian exact analogue. Actually, abelian varieties have moduli, which introduce new issues with respect to the toric case. The chapter focuses mainly on some new problems, raised by Masser, which represent a relative case of Manin–Mumford–Raynaud, where the relevant abelian variety is no longer fixed but moves in a family. The unlikely intersections of Masser's questions occur in the special case of elliptic surfaces (i.e., families of elliptic curves), and can be dealt with by a method that has been recently introduced.

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