Chow Rings, Decomposition of the Diagonal, and the Topology of Families (eBook)

Claire Voisin (Autor)

eBook Download: PDF | EPUB

2014
176 Seiten
Princeton University Press (Verlag)
978-1-4008-5053-2 (ISBN)

Infos

Claire Voisin has been a senior researcher at France's National Center for Scientific Research since 1986.

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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-Kahler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

Produkt-Details

• Erscheint lt. Verlag: 23.2.2014
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
• Themenwelt: Technik
• Schlagworte: Abelian category • Abelian varieties • abelian variety • algebraic cycle • algebraic variety • Ample line bundle • Base change • Big O notation • birational invariant • birational invariants • Bloch-Beilinson conjectures • Brauer group • CalabiЙau hypersurfaces • Canonical bundle • Cardinality • Chern class • Chow group • Chow groups • Codimension • coefficient • cohomology • Cohomology ring • collinearity • combination • Commutative diagram • complete intersection • complex algebraic varieties • Complex cobordism • Complex differential form • complex manifold • Complex torus • Complex vector bundle • coniveau • conjecture • conjugacy class • continuous function • Countable set • counterexample • cycle classes • Decomposition • decomposition isomorphism • Dense set • dense Zariski open set • derived category • diagonal • Differentiable function • differential geometry of surfaces • Dimension (vector space) • direct sum • Division by zero • Divisor • Elliptic Curve • Endomorphism • Euclidean topology • exact sequence • existential quantification • Family of curves • fiber bundle • fibration • Functoriality • generalized Bloch conjecture • generalized Hodge conjecture • Generic point • geometric coniveau • group action • Hodge classes • Hodge coniveau • Hodge conjecture • Hodge structure • Hodge structures • Hodge Theory • Homogeneous polynomial • Homology (mathematics) • hyper-Khler manifolds • hyperplane • hypersurface • Inclusion map • Integer • integral coefficients • integral cohomological decomposition • Intersection form (4-manifold) • Irreducible component • K3 surfaces • Lefschetz standard conjecture • Leray spectral sequence • line bundle • manifold • mixed Hodge structures • moduli space • Morphism • Mumford's theorem • Natural number • Natural transformation • Normal bundle • Normal function • Norm residue isomorphism theorem • Open set • parametrization • Permutation • polynomial • Presheaf (category theory) • Projection (linear algebra) • projective space • projective variety • Quadric • rational equivalence • Rational variety • Resolution of Singularities • Sheaf (mathematics) • small diagonal • smooth projective varieties • Spectral Sequence • spreading principle • SUBGROUP • Subset • Summation • Tate Conjecture • Tautological ring • tensor product • Theorem • Three-dimensional space (mathematics) • Topology • torsion coefficients • Torsor (algebraic geometry) • transcendental cohomology • Triangular Matrix • Triviality (mathematics) • unramified cohomology • Variable (mathematics) • variety • vector bundle • Zariski topology • Z-coefficients • Zero set
• ISBN-10: 1-4008-5053-3 / 1400850533
• ISBN-13: 978-1-4008-5053-2 / 9781400850532