= Normal bundle of a linear complex-projective embedding
{title2=$\nu(\mathbb{CP}^{k}\subset\mathbb{CP}^{n})\cong(\gamma_k^*)^{\oplus(n-k)}\cong\bar\gamma_k^{\oplus(n-k)}$}
Near a projective subspace, transverse directions are linear maps from the tautological line to the complementary coordinate vector space. This describes the actual complex <normal bundle>. The conjugate line and the dual line are complex-isomorphic using a <Hermitian metric>; neither is the original tautological line as a complex bundle in general.
// Target: geometry-and-topology.bigb
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