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Algebraicity of rational Hodge isometries of K3 surfaces

Abstract
Consider any rational Hodge isometry $\psi:H^2(S_1,\QQ)\rightarrow H^2(S_2,\QQ)$ between any two K\"ahler $K3$ surfaces $S_1$ and $S_2$. We prove that the cohomology class of $\psi$ in $H^{2,2}(S_1\times S_2)$ is a polynomial in Chern classes of coherent analytic sheaves over $S_1 \times S_2$. Consequently, the cohomology class of $\psi$ is algebraic whenever $S_1$ and $S_2$ are algebraic.
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