= Punctured line bundles do not determine their duality sign
For any complex line bundle $L$, a nonzero dual vector $\phi\in L_b^*$ determines the unique $v\in L_b$ with $\phi(v)=1$. In local coordinates this is fiber inversion $a\mapsto a^{-1}$, giving a smooth projection-preserving diffeomorphism between the complements of the <zero sections> of $L^*$ and $L$. It is not a fiber-linear bundle isomorphism. In particular this condition cannot distinguish the tautological line on the <complex projective line> from its dual.
Back to article page
