In Huybrechts' book Fourier-Mukai Transforms in Algebraic Geometry I found the following result due to Mukai (Page 232, Lemma 10.6)
Let $X$ and $Y$ be two $K3$ surfaces. Then the Mukai vector of any object $\mathcal{E}^{\bullet}\in D^b(X\times Y)$ is an integral cohomology class $v(\mathcal{E}^{\bullet})\in H^*(X\times Y,\mathbb{Z})$.
I want to ask if this result is for any $K3$ surfaces or just algebraic $K3$ surfaces. I am asking because the proof uses the Grothendieck-Riemann-Roch theorem (Page 127, Theorem 5.26), which holds for the algebraic case only.
Moreover, I am not sure if $X\times Y$ has the resolution property. If not, then it is not even easy to define the Chern character of a general object $\mathcal{E}^{\bullet}\in D^b(X\times Y)$.
