Let $X$ be a smooth projective variety over a field $k$ with char$k=0$ and $\mathcal{L}$ be a very ample line bundle on $X$. Let $\mathcal{F}$ be a coherent sheaf on $X$. It is well-know that if $\mathcal{F}\otimes \mathcal{L}\cong \mathcal{F}$, then $\text{dim}(\text{supp}\mathcal{F})=0$. Actually it can be proved by looking at the degree of the Hilbert polynomial associated with $\mathcal{F}$.

Now Let $X$ and $Y$ be two smooth projective varieties over a field $k$ with char$k=0$ and $\mathcal{L}$ be a very ample line bundle on $X$. Let $p: X\times Y\to X$ be the projection. Let $\mathcal{F}$ be a coherent sheaf on $X\times Y$ and we consider $p(\text{supp}\mathcal{F})$, the projection to $X$ of the support of $\mathcal{F}$. If $\mathcal{F}\otimes p^*\mathcal{L}\cong \mathcal{F}$, do we have $$\text{dim}(p(\text{supp}\mathcal{F}))=0?$$

By studying the Hilbert polynomial it can be proved that $\text{dim}(\text{supp}R^ip_*\mathcal{F})=0$. But this seems to be weaker than the result I expect.