Particularly interested in coherent sheaves on projective varieties or complex manifolds.
2026-03-25 01:18:11.1774401491
Can a coherent sheaf with no global sections have non-trivial higher cohomology?
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Yes. On $\mathbb{P}^1_\mathbb{C}$, the line bundle $\mathcal{O}(-2)$ has no global sections but $H^1(\mathcal{O}(-2))\cong \mathbb{C}$.