Weak Lefschetz for Chow Groups: Infinitesimal Lifting
نویسندگان
چکیده
Let X be a smooth projective variety over an algebraically closed field k of characteristic zero and Y ⊂ X a smooth ample hyperplane section. The Weak Lefschetz conjecture for Chow groups states that the natural restriction map CH(X)Q → CH(Y )Q is an isomorphism for all p < dim(Y )/2. In this note, we revisit a strategy introduced by Grothendieck to attack this problem by using the Bloch-Quillen formula to factor this morphism through a continuous K-cohomology group on the formal completion of X along Y . This splits the conjecture into two smaller conjectures: one consisting of an algebraization problem and the other dealing with infinitesimal liftings of algebraic cycles. We give a complete proof of the infinitesimal part of the conjecture.
منابع مشابه
Towards Connectivity for Codimension 2 Cycles: Infinitesimal Deformations
Let X be a smooth projective variety over an algebraically closed field k ⊂ C of characteristic zero, and Y ⊂ X a smooth complete intersection. The Weak Lefschetz theorem states that the natural restriction map H(X(C), Q) → H(Y (C), Q) on singular cohomology is an isomorphism for all i < dim(Y ). The Bloch-Beilinson conjectures on the existence of certain filtrations on Chow groups combined wit...
متن کاملAn Infinitesimal Noether-lefschetz Theorem for Chow Groups
Let X be a smooth, complex projective variety, and Y be a very general, sufficiently ample hypersurface in X. A conjecture of M. V. Nori states that the natural restriction map CH(X)Q → CH(Y )Q is an isomorphism for all p < dimY and an injection for p = dimY . This is the generalized Noether-Lefschetz conjecture. We prove an infinitesimal version of this conjecture.
متن کاملLefschetz Decompositions for Quotient Varieties
In an earlier paper, the authors constructed an explicit Chow Kunneth decomposition for quotient varieties of Abelian varieties by actions of finite groups. In the present paper, the authors extend the techniques there to obtain an explicit Lefschetz decomposition for such quotient varieties for the Chow-Kunneth projectors constructed there.
متن کاملLefschetz Theorems for Torsion Algebraic Cycles in Codimension 2
Let Y be a smooth projective variety over C, and X be a smooth hypersurface in Y . We prove that the natural restriction map on Chow groups of codimension two cycles is an isomorphism when restricted to the torsion subgroups provided dimY ě 5. We prove an analogous statement for a very general hypersurface X Ă P of degree ě 5. In the more general setting of a very general hypersurface X of suff...
متن کاملA general construction of partial Grothendieck transformations
Fulton and MacPherson introduced the notion of bivariant theories related to Riemann-Roch-theorems, especially in the context of singular spaces. This is powerful formalism, which is a simultaneous generalization of a pair of contravariant and covariant theories. Natural transformations of bivariant theories are called Grothendieck transformations, and these generalize a pair of ordinary natura...
متن کامل