Numerical and Homological Equivalence of Algebraic Cycles on Hodge Manifolds

Abstract

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Algebraic cycles on the relative symmetric powers and on the relative Jacobian of a family of curves. I

“…It is well known that the standard conjecture B of Lefschetz type for X and for a projective bundle over X are equivalent (see [18]). Also, if it is true for some variety X then it is true for an ample divisor in X.…”

On the interior motive of certain Shimura varieties: the case of Picard surfaces

“…7.1.2]. But according to [Li,Thm. 4], numerical and homological equivalence coincide on Abelian varieties over C. In other words,…”

Grothendieck standard conjectures, morphic cohomology and Hodge index theorem

“…The Grothendieck standard conjecture B (GSCB for short) says that the adjoint operator Λ is algebraic, i.e., there is a cycle β on X × X such that Λ : H * (X; Q) −→ H * (X; Q) is got by lifting a class from X to X × X by the first projection, cupping with β and taking the image in H * (X; Q) by the Gysin homomorphism associated to the second projection. For abelian varieties, the GSCB was proved by Lieberman in [19], and we know the GSCB for a smooth variety which is a complete intersection in some projective space and for Grassmannians (see [11]). …”