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In this article we generalize the injectivity of the push-forwardhomomorphism at the level of Chow groups, induced by the closed embedding of$\Sym^m C$ into $\Sym^n C$ for $m\leq n$, where $C$ is a smooth projectivecurve, to symmetric powers of a smooth projective variety of higher dimension.We also prove the analog of this theorem for product of symmetric powers ofsmooth projective varieties. As an application we prove the injectivity of thepush-forward homomorphism at the level of Chow groups, induced by the closedembedding of self product of theta divisor into the self product of theJacobian of a smooth projective curve.

On the closed embedding of the product of theta divisors into product of Jacobians and Chow groups