For the question, everything is over an algebraically closed field and by a scheme we mean a scheme of finite type. The theorem of the cube is the following:
Let $X$ be a complete variety, $Y$ a scheme, and $L$ a line bundle on $X\times Y$. Then there is a unique closed subscheme $Y_1$ of $Y$ satisfying the following properties: (a) If $L_1$ is the restriction of $L$ to $X\times Y_1$, there is a line bundle $M_1$ on $Y_1$ and an isomorphism $p_2^*M_1\rightarrow L_1$ on $X\times Y_1$. (b), since (b) has little to do with the question, let us focus on (a). I am puzzled by the following thing: by the Künneth formula, if $p_2^*M_1\rightarrow L_1$ is an isomorphism, then $M_1 \rightarrow {p_2}_*(L_1)$ is an isomorphism. Does anyone knows what is the content of the Künneth formula mentioned here and more importantly, why the statement is true? Thanks!