Which is the correct universal line bundle: the tautological bundle or its dual?

Question

With topological line bundles over $\mathbb{C}$, one learns that every line bundle is a pullback of the universal line bundle, which is the tautological line bundle over $\mathbb{C}P^\infty.$

In complex geometry, one learns that every holomorphic line bundle with enough global sections that don't simultaneously vanish is a pullback of $\mathcal{O}(1)$, which is the dual of the tautological line bundle over $\mathbb{P}_N$ for some large $N$.

In general, unlike for real bundles, a complex vector bundle is not isomorphic to its dual, but rather to the complex conjugate is isomorphic to its dual. How can I harmonize these two statements? Is it that when we view the tautological line bundle over $\mathbb{P}_N$ as topological bundle, forgetting its holomorphic structure, we're allowing non-holomorphic sections? It seems like the holomorphic statement is the more refined result. Is there some way to see that the dual of the tautological line bundle is the more appropriate universal bundle in the topological case?

Answer 1

I disagree that the dual of the tautological line bundle is the more appropriate universal bundle in the topological case. The point of taking duals in the holomorphic setting is that holomorphic sections of powers of the dual bundle, namely homogeneous polynomials, are "functions" of lines, so the relationship between these sections and lines is fundamentally contravariant.

Working with the tautological bundle means you care directly about lines whereas working with the dual means you care about functions of lines. To see the same idea in a less complicated setting, consider the difference between caring about a vector space $V$ and caring about the algebra $S(V^{\ast})$ of polynomial functions on $V$, which is built from symmetric powers of the dual vector space.