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Linear ind-Grassmannians

We consider ind-varieties obtained as direct limits of chains of embeddings $X_1\stackrel{ϕ_1}{\hookrightarrow}\dots\stackrel{ϕ_{m-1}}{\hookrightarrow} X_m\stackrel{ϕ_m}{\hookrightarrow}X_{m+1}\stackrel{ϕ_{m+1}}{\hookrightarrow}\dots$, where each $X_m$ is a Grassmannian or an isotropic Grassmannian (possibly mixing Grassmannians and isotropic Grassmannians), and the embeddings $ϕ_m$ are linear in the sense that they induce isomorphisms of Picard groups. We prove that any such ind-variety is isomorphic to one of certain standard ind-Grassmannians and that the latter are pairwise non-isomorphic ind-varieties.