A <holomorphic line bundle> is <ample> when some positive <tensor power> is <very ample>, so its sections define an embedding into <Complex projective space>. It is a <positive holomorphic line bundle> when it has a <Hermitian metric> with positive Chern curvature $iF_\nabla$. The <Kodaira embedding theorem> says that a compact <complex manifold> carrying a positive holomorphic line bundle is projective; sufficiently high tensor powers give a <holomorphic embedding> into projective space.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 / 3 / c / Solution

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The Kodaira embedding theorem says that a compact <complex manifold> carrying a <positive holomorphic line bundle> is projective: sufficiently large <tensor powers> of the bundle have global holomorphic sections defining a <holomorphic embedding> into <Complex projective space>. Conversely, the restriction of $\mathcal O(1)$ gives a positive line bundle on every projective complex manifold.

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 Incoming links: Kodaira embedding theorem