Normal bundle of a linear complex-projective embedding ($\nu ({\mathbb{CP}}^k\subset {\mathbb{CP}}^n)\cong ({\gamma }_k^{\ast }{)}^{\oplus (n-k)}\cong {\bar{\gamma }}_k^{\oplus (n-k)}$)

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