Introduction

2017 ◽  
Author(s):  
Claire Voisin
Keyword(s):  
Chow Groups ◽  
Projective Varieties ◽  
General Fiber ◽  
Main Ideas ◽  
Smooth Projective Varieties

This chapter discusses some crucial notions to the interplay between cohomology and Chow groups, and also to the consequences, for the topology of a family of smooth projective varieties, of statements concerning Chow groups of the general or very general fiber. It surveys the main ideas and results presented throughout this volume. First, the chapter discusses the decomposition of the diagonal and spread. It then explains the generalized Bloch conjecture, the converse to the generalized decomposition of the diagonal. Next, the chapter turns to the decomposition of the small diagonal and its application to the topology of families. Finally, the chapter discusses integral coefficients and birational invariants before providing a brief overview of the following chapters.

scholarly journals Representability of Chow groups of codimension three cycles

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Kalyan Banerjee
Keyword(s):  
Chow Groups ◽  
Generic Fiber ◽  
Projective Varieties ◽  
Rational Equivalence ◽  
Finite Dimensional ◽  
Etale Cohomology ◽  
Étale Cohomology ◽  
Smooth Projective Varieties

Abstract Assume that we have a fibration of smooth projective varieties X → S over a surface S such that X is of dimension four and that the geometric generic fiber has finite-dimensional motive and the first étale cohomology of the geometric generic fiber with respect to ℚ l coefficients is zero and the second étale cohomology is spanned by divisors. We prove that then A 3(X), the group of codimension three algebraically trivial cycles modulo rational equivalence, is dominated by finitely many copies of A 0(S); this means that there exist finitely many correspondences Γi on S × X such that Σ i Γi is surjective from A 2(S) to A 3(X).

scholarly journals On the closed embedding of the product of theta divisors into product of Jacobians and Chow groups

2018 ◽  
Vol 29 (03) ◽  
pp. 1850021
Author(s):  
Kalyan Banerjee
Keyword(s):  
Smooth Projective Variety ◽  
Chow Groups ◽  
Projective Curve ◽  
Theta Divisor ◽  
Projective Varieties ◽  
Smooth Projective Curve ◽  
Symmetric Powers ◽  
Closed Embedding ◽  
Push Forward ◽  
Smooth Projective Varieties

In this paper, we generalize the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of [Formula: see text] into [Formula: see text] for [Formula: see text], where [Formula: see text] is a smooth projective curve, to symmetric powers of a smooth projective variety of higher dimension. We also prove the analog of this theorem for product of symmetric powers of smooth projective varieties. As an application we prove the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of self-product of theta divisor into the self-product of the Jacobian of a smooth projective curve.

On the Zariski-van Kampen Theorem

2003 ◽  
Vol 55 (1) ◽  
pp. 133-156 ◽  
Author(s):  
Ichiro Shimada
Keyword(s):  
Projective Varieties ◽  
General Fiber

AbstractLet f : E → B be a dominant morphism, where E and B are smooth irreducible complex quasi-projective varieties, and let Fb be the general fiber of f. We present conditions under which the homomorphism π1(Fb) → π1(E) induced by the inclusion is injective.

scholarly journals Schwarzenberger bundles on smooth projective varieties

2016 ◽  
Vol 220 (9) ◽  
pp. 3307-3326 ◽  
Author(s):  
Enrique Arrondo ◽  
Simone Marchesi ◽  
Helena Soares
Keyword(s):  
Projective Varieties ◽  
Smooth Projective Varieties

scholarly journals Abelian Varieties as Automorphism Groups of Smooth Projective Varieties

2018 ◽  
Vol 2020 (7) ◽  
pp. 1942-1956
Author(s):  
Davide Lombardo ◽  
Andrea Maffei
Keyword(s):  
Automorphism Group ◽  
Projective Variety ◽  
Automorphism Groups ◽  
Abelian Varieties ◽  
Smooth Projective Variety ◽  
Projective Varieties ◽  
Smooth Projective Varieties

Abstract We determine which complex abelian varieties can be realized as the automorphism group of a smooth projective variety.

scholarly journals Growth rate of ample heights and the Dynamical Mordell–Lang conjecture

2018 ◽  
Vol 14 (10) ◽  
pp. 2673-2685
Author(s):  
Kaoru Sano
Keyword(s):  
Growth Rate ◽  
Explicit Formula ◽  
Number Fields ◽  
Rational Points ◽  
Positive Answer ◽  
Projective Varieties ◽  
Smooth Projective Varieties

We provide an explicit formula on the growth rate of ample heights of rational points under iteration of endomorphisms of smooth projective varieties over number fields. As an application, we give a positive answer to a variant of the Dynamical Mordell–Lang conjecture for pairs of étale endomorphisms, which is also a variant of the original one stated by Bell, Ghioca, and Tucker in their monograph.

scholarly journals A Riemann–Hurwitz Theorem for the Algebraic Euler Characteristic

2017 ◽  
Vol 60 (3) ◽  
pp. 490-509
Author(s):  
Andrew Fiori
Keyword(s):  
Euler Characteristic ◽  
Euler Characteristics ◽  
Projective Varieties ◽  
Coherent Sheaves ◽  
Irreducible Components ◽  
Arbitrary Dimensions ◽  
Smooth Projective Varieties ◽  
Ramification Locus ◽  
Hurwitz Theorem

AbstractWe prove an analogue of the Riemann–Hurwitz theorem for computing Euler characteristics of pullbacks of coherent sheaves through finite maps of smooth projective varieties in arbitrary dimensions, subject only to the condition that the irreducible components of the branch and ramification locus have simple normal crossings.

A GEOMETRIC VERSION OF DYSON'S LEMMA FOR FACTORS OF ARBITRARY DIMENSION

2002 ◽  
Vol 13 (01) ◽  
pp. 43-65 ◽  
Author(s):  
MARKUS WESSLER
Keyword(s):  
Algebraic Geometry ◽  
Arbitrary Dimension ◽  
Product Theorem ◽  
Projective Varieties ◽  
Quantitative Version ◽  
Weak Positivity ◽  
Projective Lines ◽  
Smooth Projective Varieties ◽  
Geometric Analogue

This paper generalizes the geometric part of the Esnault–Viehweg paper on Dyson's Lemma for a product of projective lines. Using the method of weak positivity from algebraic geometry, we are able to study products of smooth projective varieties of arbitrary dimension and to prove a geometric analogue of Dyson's Lemma for this case. Our main result is in fact a quantitative version of Faltings' product theorem.

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