Does it make sense to define a holomorphic structure on $\mathbb{C}P^\infty$ and vector bundles over it? Let $E \to \mathbb{C}P^\infty$ be any topological complex vector bundle over the infinite complex projective space. I'm wondering if it makes sense to possibly define a "holomorphic structure" on $E$. This a priori requires a complex structure on $\mathbb{C}P^\infty$, which is also something I don't know whether it exists or is well-defined, given that we're working with an infinite dimensional manifold. But it feels natural that there should be at least some notion of holomorphicity on the tautological line bundle over $\mathbb{C}P^\infty$. Is there anything in the literature about this?
 A: Yes, there is lots of literature on this subject.
However, Tyurin proved that all vector bundles on $CP^\infty$ are
direct sum of line bundles. There are several more recent papers by
Penkov and Tikhomirov about vector bundles on $C P^\infty$ (they treat other infinite-dimensional
manifolds, too).

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*A. N. Tjurin (Tyurin), Vector bundles of finite rank over infinite varieties, Izv. Akad. Nauk SSSR Ser. Mat. 40:6 (1976), 1248–1268; English transl. in Math. USSR-Izv. 10:6 (1976), 1187–1204. https://doi.org/10.1070/IM1976v010n06ABEH001832


*Penkov, I. B. (D-JACOB); Tikhomirov, A. S. (RS-HSEM)
On the Barth–Van de Ven–Tyurin-Sato theorem, (Russian)
Mat. Sb. 206 (2015), no. 6, 49–84; translation in
Sb. Math. 206 (2015), no. 5-6, 814–848 https://doi.org/10.1070/SM2015v206n06ABEH004480, https://arxiv.org/abs/1405.3897


*Penkov, I. B.; Tikhomirov, A. S. Triviality of vector bundles on twisted ind-Grassmannians. (Russian) Mat. Sb. 202 (2011), no. 1, 65–104; translation in Sb. Math. 202 (2011), no. 1-2, 61–99, https://doi.org/10.1070/SM2011v202n01ABEH004138, https://arxiv.org/abs/0706.3912
