Chow Rings, Decomposition of the Diagonal, and the Topology of Families

Chow Rings, Decomposition of the Diagonal, and the Topology of Families

Claire Voisin
Copyright Date: 2014
Pages: 176
https://www.jstor.org/stable/j.ctt5hhp7w
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    Chow Rings, Decomposition of the Diagonal, and the Topology of Families
    Book Description:

    In this book, Claire Voisin 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 volume 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 Voisin. The book 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 Voisin 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.

    eISBN: 978-1-4008-5053-2
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-vi)
  3. Preface
    (pp. vii-viii)
    Claire Voisin
  4. Chapter One Introduction
    (pp. 1-14)

    These lectures are devoted 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.

    A crucial notion is that of theconiveauof a cohomology. A Betti cohomology class has geometric coniveau ≥cif it is supported on a closed algebraic subset of codimension ≥c. The coniveau of a class of degreekis$\le \frac{k}{2}$. As a smooth projective varietyXhas nonzero cohomology in degrees 0, 2, 4, … obtained by...

  5. Chapter Two Review of Hodge theory and algebraic cycles
    (pp. 15-35)

    We follow [43] and [101, II]. LetXbe a scheme over a fieldK(which in practice will always be a quasi-projective scheme, that is, a Zariski open set in a projective scheme). Let${{\cal Z}_{k}}(X)$be the group ofk-dimensional algebraic cycles ofX, that is, the free abelian group generated by the (reduced and irreducible) closedk-dimensional subvarieties ofXdefined overK. IfYXis a subscheme of dimension ≤k, we can associate a cycle$c(Y)\in {{\cal Z}_{k}}(X)$to it as follows: Set\[c(Y)=\sum\limits_{W}{{{n}_{W}}}W, \caption {(2.1)}\]where the sum is taken over thek-dimensional irreducible reduced components of...

  6. Chapter Three Decomposition of the diagonal
    (pp. 36-54)

    The following result is essentially due to Bloch and Srinivas [15]. Letf:XYbe a smooth projective morphism, whereXandYare smooth (for simplicity), and letZ∈ CHk(X) be a cycle inX. Consider the following property.

    (∗)There exists a subvariety X′X such that for every yY, the cycle${{Z}_{y}}:={{Z}_{\left| {{X}_{y}} \right.}}:=j_{y}^{*}Z\in \text{C}{{\text{H}}^{k}}({{X}_{y}})$vanishes in$\text{C}{{\text{H}}^{k}}({{X}_{y}}-X_{y}^{\prime})$,where jyis the inclusion of the fiber Xy=f−1(y)into X.

    Theorem 3.1.If Z satisfies the property(∗),then there exist an integer m> 0and a cycle Z′ supported in X′,...

  7. Chapter Four Chow groups of large coniveau complete intersections
    (pp. 55-87)

    Consider a smooth complete intersection$X\subset {{\mathbb{P}}^{n}}$ofrhypersurfaces of degree${{d}_{1}}\le \cdots \le {{d}_{r}}$. By the Lefschetz hyperplane section theorem (see [101, II, 1.2.2]), the only interesting Hodge structure in the cohomology ofXis the Hodge structure on$H_{B}^{n-r}(X,\mathbb{Q})$, and in fact on the primitive part of it (that is, the orthogonal of the restriction of$H_{B}^{*}({{\mathbb{P}}^{n}},\mathbb{Q})$with respect to the intersection pairing). We will say thatXhas Hodge coniveaucif the Hodge structure on$H_{B}^{n-r}{{(X,\mathbb{Q})}_{\text{prim}}}$has coniveauc.

    The Hodge coniveau of a complete intersection in projective space is computed as follows (see [48] for the...

  8. Chapter Five On the Chow ring of K3 surfaces and hyper-Kähler manifolds
    (pp. 88-122)

    The following theorem is proved in [11].

    Theorem 5.1 (Beauville and Voisin 2004).Let S be a K3surface, Di∈ CH1(S)be divisors on S and nijbe integers. Then if the0-cycle${{\sum }_{i,j}}{{n}_{ij}}{{D}_{i}}{{D}_{j}}\in \text{C}{{\text{H}}_{0}}(S)$is cohomologous to0on S, it is equal to0inCH0(S).

    Proof. It suffices to prove that there is a 0-cycleoof degree 1, with the property that for any two line bundlesL,L′ onS,\[L\cdot {L}'=\deg (L\cdot {L}')\ o\ \text{in}\quad \text{C}{{\text{H}}_{0}}(S). \caption {(5.1)}\]

    The cycleois defined to be the class of any point ofScontained in a (singular) rational curveCS. We claim...

  9. Chapter Six Integral coefficients
    (pp. 123-154)

    Atiyah and Hirzebruch [5] found counterexamples to the Hodge conjecture (Conjecture 2.25) stated for degree 2k integralHodge classes (as opposed torationalHodge classes) whenk≥ 2. In degree 2, the most optimistic statements are true, due to the Lefschetz theorem on (1, 1)-classes.

    In [95], Totaro revisited the examples of Atiyah and Hirzebruch and reformulated more directly the obstructions they had found, using the complex cobordism graded ring MU* (X) ofX. Let us first describe this ring that is defined for all differentiable compact manifolds: Given such anX, we consider first of all the objects...

  10. Bibliography
    (pp. 155-162)
  11. Index
    (pp. 163-163)