#### Room 326

#### Rational equivalence of algebraic cycles supported on a general hyperplane section

#### Kalyan Banerjee

##### IMSc, Chennai

*Let X be a smooth projective variety of even dimension 2p and we fix an embedding of X into some projective space P^N. Let Y be a smooth hyperplane section of X and let j denote the closed embedding of Y into X. Then j induces the push-forward homomorphism j_* from A^p(Y) to A^{p+1}(X). In this talk we will be interested in understanding the kernel of j_*. We prove that, if A^p(Y) is isomorphic to an abelian variety A(Y), then the kernel of j_* is a countable union of translates of an abelian subvariety A_0(Y) of A(Y). Moreover for a very general hyperplane section Y, we prove that, this A_0(Y) is either {0} or A(Y). Furthermore if we have certain conditions on A^p(Y) and A^{p+1}(X) then for a very general hyperplane section Y, A_0(Y) is actually {0}, proving that, for a very general hyperplane section Y of X, the kernel of j_* is actually countable. We show that, this is the case when X is a K3 surface or a smooth cubic fourfold inside P^5.*

Done