Every rational Hodge isometry between two K3 surfaces is algebraic

We prove that given any rational Hodge isometry $\psi:H^2(S_1,\mathbb{Q})\rightarrow H^2(S_2,\mathbb{Q})$ between any two K\"ahler $K3$ surfaces $S_1$ and $S_2$ 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.