On Certain Numerical Invariants of Algebraic Varieties with Application to Abelian Varieties

1921 ◽  
Vol 22 (4) ◽  
pp. 407 ◽  
Author(s):  
Solomon Lefschetz
Keyword(s):  
Abelian Varieties ◽  
Algebraic Varieties ◽  
Numerical Invariants

Local Bounds for Torsion Points on Abelian Varieties

2008 ◽  
Vol 60 (3) ◽  
pp. 532-555 ◽  
Author(s):  
Pete L. Clark ◽  
Xavier Xarles
Keyword(s):  
Abelian Varieties ◽  
Residue Field ◽  
Rational Numbers ◽  
Abelian Surface ◽  
Torsion Subgroup ◽  
Good Reduction ◽  
Torsion Points ◽  
Ramification Index ◽  
Numerical Invariants ◽  
Split Torus

AbstractWe say that an abelian variety over a p-adic field K has anisotropic reduction (AR) if the special fiber of its Néronminimal model does not contain a nontrivial split torus. This includes all abelian varieties with potentially good reduction and, in particular, those with complex or quaternionic multiplication. We give a bound for the size of the K-rational torsion subgroup of a g-dimensional AR variety depending only on g and the numerical invariants of K (the absolute ramification index and the cardinality of the residue field). Applying these bounds to abelian varieties over a number field with everywhere locally anisotropic reduction, we get bounds which, as a function of g, are close to optimal. In particular, we determine the possible cardinalities of the torsion subgroup of an AR abelian surface over the rational numbers, up to a set of 11 values which are not known to occur. The largest such value is 72.

scholarly journals Fixed point theorems involving numerical invariants

2019 ◽  
Vol 155 (2) ◽  
pp. 260-288
Author(s):  
Olivier Haution
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Fixed Point Theorems ◽  
Algebraic Varieties ◽  
Numerical Invariants

We exhibit invariants of smooth projective algebraic varieties with integer values, whose nonvanishing modulo $p$ prevents the existence of an action without fixed points of certain finite $p$-groups. The case of base fields of characteristic $p$ is included. Counterexamples are systematically provided to test the sharpness of our results.

Projective characters and invariants of algebraic varieties

1937 ◽  
Vol 33 (2) ◽  
pp. 188-198
Author(s):  
L. Roth
Keyword(s):  
General Type ◽  
A Priori ◽  
Direct Calculation ◽  
Linear Functions ◽  
Algebraic Varieties ◽  
Arithmetic Genus ◽  
Canonical Systems ◽  
Numerical Invariants ◽  
Simple Notation ◽  
Relative Invariants

It is a familiar fact that the arithmetic genus pa and the arithmetic linear genus ω of a general surface are linear functions of its four projective characters; and we find by direct calculation that a similar property holds for the numerical invariants of a general threefold. The question thus arises, whether this result can be established a priori for any algebraic variety Vk of general type, since in that case we should have a simple means of determining its numerical invariants. It has been shown by Severi that, subject to a certain assumption, the arithmetic genus pk of Vk is a function of its projective characters, while it is known that, for k ≤ 4, pk coincides with the arithmetic genus Pa obtained by the second definition (§ 5). In the present paper we obtain, by using Severi's postulate, expressions for the arithmetic genera of a V3 and a V4 in terms of their projective characters. We obtain also the characters of their virtual canonical systems and hence derive formulae for the relative invariants Ωi. For this purpose we replace certain projective characters of Vk by others which are more easily computed and better adapted to a simple notation.

Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

2017 ◽  
Author(s):  
Claire Voisin
Keyword(s):  
Recent Work ◽  
Chow Groups ◽  
Ring Structure ◽  
Complete Intersections ◽  
Algebraic Varieties ◽  
Chow Ring ◽  
Hodge Structures ◽  
Chow Rings ◽  
Geometric Methods ◽  
The Right

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

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