Remarks on Zero-Cycles of Self-Products of Varieties
Claire Voisin · 2023
1.1 This paper proposes a few results and rises many questions on algebraic cycles on selfproducts X k of a variety X . If X is a curve, it is well known that CH 0 ( X k ) = CH 0 ( X (k +1) ) for k ≥ g ( X ) where X (k) denotes the k th symmetric product of X , and the isomorphism is given by the map μ x0 : X (k) → X (k+1) , μ x 0 ( z ) = z + x 0 , for any point x 0 ∈ X . Correspondingly, one has: https://www.w3.org/1998/Math/MathML" display="inline"> ∀ k ≥ g ( X ) , ∀ l ∈ ℕ , H 0 ( Ω X ( k + 1 ) l ) = H 0 ( Ω X ( k ) l ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419983/68c56c13-fae3-4854-a9aa-70d7297a8ed4/content/ieq2245.tif "/> , which is the effect on holomorphic forms of the previous equality, using Mumford-Roitman theorem ([ 16 ], [ 22 ]). This last fact generalizes to higher dimensional varieties as follows: Assume https://www.w3.org/1998/Math/MathML" display="inline"> H 0 ( Ω X l ) = 0 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419983/68c56c13-fae3-4854-a9aa-70d7297a8ed4/content/ieq2246.tif "/> for l even, l ≠ 0; then for https://www.w3.org/1998/Math/MathML" display="inline"> k ≥ ∑ i = 1 dim X h i , 0 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419983/68c56c13-fae3-4854-a9aa-70d7297a8ed4/content/ieq2247.tif "/> and any https://www.w3.org/1998/Math/MathML" display="inline"> l , H 0 ( Ω X k l ) S k = H 0 ( Ω X k + 1 l ) E k + 1 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419983/68c56c13-fae3-4854-a9aa-70d7297a8ed4/content/ieq2248.tif "/> , where () ∈ means the invariant part under the action of the symmetric group 5.. According to Bloch-Beilinson conjectures [ 11 ], this should imply that https://www.w3.org/1998/Math/MathML" display="inline"> ( M I ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419983/68c56c13-fae3-4854-a9aa-70d7297a8ed4/content/ieq2249.tif "/> for https://www.w3.org/1998/Math/MathML" display="inline"> k ≥ ∑ i = 1 dim X h i , 0 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419983/68c56c13-fae3-4854-a9aa-70d7297a8ed4/content/ieq2250.tif "/> . I have no general result on this but I will construct in section 3 families of threefolds with no H 2,0, H 1,0 and which satisfy: (MI) for k ≥ h 3,0 (X)