On the support of admissible subcategories

Let $X$ be a smooth proper variety over an algebraically closed field of characteristic zero, and let $\mathcal{A}\subset D_{\mathrm{c}\mathrm{o}\mathrm{h}}^b(X)$ be an admissible subcategory. Let $Z\subset X$ be the union of set-theoretical supports of all objects in $\mathcal{A}$ and assume that $Z\mathrm{\ne }X$. We show that for any morphism from $Z$ to an abelian variety each fiber has no isolated points; this implies, for example, that $Z$ cannot be isomorphic to an abelian variety. The key input is the fact that while not all line bundles on $Z$ lift to infinitesimal thickenings of $Z$, sufficiently many do: specifically, we show that for any infinitesimal thickening $Z\subset \tilde{Z}$ the restriction morphism ${\mathrm{P}\mathrm{i}\mathrm{c}}^0(\tilde{Z})\to {\mathrm{P}\mathrm{i}\mathrm{c}}^0(Z)$ on the connected components of Picard schemes induces an isogeny between Albanese group schemes of those connected components.