ag.algebraic geometry - Algebraic cycles of dimension 2 on the square of a generic abelian surface

Here is an easy $5$ dimensional space of cycles: Inside $A times A$, consider the subvarieties ${ (a,b) : a=mb }$, for $m=0$, $1$, $2$, $3$, $4$. I will show that these are linearly independent over $mathbb{Q}$.

By Kunneth and Poincare,

$$H^4(A times A, mathbb{Q}) cong bigoplus_{i=0}^4 H^{i}(A, mathbb{Q}) otimes H^{4-i}(A, mathbb{Q}) cong bigoplus_{i=0}^4 mathrm{End}(H^{i}(A, mathbb{Q})).$$

The graph of multiplication by $m$, in this presentation, has class

$$(mathrm{Id}, m mathrm{Id}, m^2 mathrm{Id}, m^3 mathrm{Id}, m^4 mathrm{Id})$$

Since the Vandermonde matrix

$$begin{pmatrix} 0^0 & 0^1 & 0^2 & 0^3 & 0^4 \ 1^0 & 1^1 & 1^2 & 1^3 & 1^4 \ 2^0 & 2^1 & 2^2 & 2^3 & 2^4 \ 3^0 & 3^1 & 3^2 & 3^3 & 3^4 \ 4^0 & 4^1 & 4^2 & 4^3 & 4^4 end{pmatrix}$$
has nonzero determinant, the $5$ classes I listed are linearly independent over $mathbb{Q}$.

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