Chow Rings, Decomposition of the Diagonal, and the Topology of Families
Claire Voisin
Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Beschreibung
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.
Kundenbewertungen
Collinearity, Countable set, Cohomology, Line bundle, Sheaf (mathematics), Codimension, Abelian category, Abelian variety, Projective variety, Topology, Complete intersection, Normal bundle, Theorem, Natural transformation, Conjugacy class, Projective space, Zariski topology, Fiber bundle, Subgroup, Hypersurface, Open set, Torsor (algebraic geometry), Elliptic curve, Irreducible component, Morphism, Norm residue isomorphism theorem, Dimension (vector space), Integer, Complex differential form, Permutation, Resolution of singularities, Variable (mathematics), Algebraic variety, Divisor, Zero set, Dense set, Parametrization, Chow group, Division by zero, Differential geometry of surfaces, Intersection form (4-manifold), Algebraic cycle, Direct sum, Moduli space, Exact sequence, Conjecture, Commutative diagram, Generic point, Derived category, Ample line bundle, Fibration, Existential quantification, Summation, Tensor product, Coefficient, Cohomology ring, Rational variety, Hodge conjecture, Quadric, Birational invariant, Diagonal, Hodge theory, Hodge structure, Differentiable function, Leray spectral sequence, Vector bundle, Polynomial, Subset, Complex cobordism, Homology (mathematics)