Hartogs extension theorem ($\mathcal{O}(B\setminus \{0\})=\mathcal{O}(B)$ for ${\dim }_{\mathbb{C}}B\ge 2$)

In complex dimension at least two, a holomorphic function on a punctured coordinate ball extends holomorphically through its centre. Componentwise extension in a holomorphic local frame gives the same statement for holomorphic sections. This preserves global sections after removing finitely many points, but does not imply preservation of higher sheaf cohomology. This analytic theorem differs from the set-theoretic Hartogs theorem.