Let $Y$ be the smooth manifold underlying a K3 surface. As a manifold, $Y$ is diffeomorphic to $\{[x_0:x_1:x_2:x_3]\in\mathbb{C}P^3\colon X_0^4+x_1^4+X_2^4+X_3^4=1\}$. It is well known that $H^2(Y,\mathbb{Z})\cong U^{\oplus 3}\oplus(-E_8)^{\oplus 2}\cong \mathbb{Z}^{\oplus 22}$.
From the classification results of principal $S^1$-bundles $p:M\to Y$, we have know that they are classified by $H^2(Y,\mathbb{Z})$. Namely, by picking a connection 1-form $\theta$ on $M$, which satisfies $d\theta=p^*\omega$ for some closed $\omega\in \Omega^2(Y)$, we get a map $$\Phi:\{\text{Isomorphism classes of principal $S^1$-bundles $p:M\to Y$}\}\to H^2(Y,\mathbb{Z}),$$
by sending $[p:M\to Y]$ to $[\omega]$.
In the case that $Y$ is a K3 surface, I am wondering if there are concrete presentations of these principal $S^1$-bundles around. More concretely, suppose we have have a basis $c_1,\cdots,c_{22}$ of $H^2(Y,\mathbb{Z})$, can we construct 22 principal $S^1$-bundles with classes $c_1,\cdots,c_{22}$.
Since $S^1\to S^7\to \mathbb{C}P^3$ is an $S^1$-bundle, I tried to realize $Y$ as an $S^1$-bundle $S^1\to M\to Y$, where $M\subset S^7$ is some smooth submanifold. But this is probably not going to give me all bundles over $Y$, so any help or any link to literature would be helpful!
Best Answer
You can say a fair amount about the topology of the total spaces of the different bundles, although I suspect none of them is a particularly well-known manifold that has a `name'. (Except of course for the trivial bundle; I guess you could say $S^1 \times$ a K3 surface is a well-known manifold.)
The main observation is that you can read off the fundamental group and the basic homological invariants of $M$ (using your notation) in terms of the Euler class $e \in H^2(Y)$. It seems that the fundamental group is cyclic of order the divisibility of $e$. For the cohomology (and hence homology by duality) you can use the Gysin sequence for the cohomology
$\cdots \to H^j(M) \to H^{j-1}(Y) \stackrel{\cup e}{\rightarrow} H^{j+1}(Y) \to H^{j+1}(M) \to \cdots $
To go further, you'd need to know more about the automorphism group of the intersection form of $Y$. I think (but I'm not certain) that the automorphism group is transitive on element of a given square and divisibility. Assuming this to be the case, I believe that implies that any two Euler classes with the same square and divisibility are related by a self-diffeomorphism of $Y$. That certainly lessens the complexity of the problem.
For the automorphism group, I would look at Wall's On the orthogonal groups of unimodular quadratic forms. II. J. Reine Angew. Math. 213 (1963/64), 122–136. For realizing automorphisms by diffeomorphisms, see T. Matumoto, On diffeomorphisms of a K3 surface, Algebraic and topological theories (Kinosaki, 1984), Kinokuniya, Tokyo, 1986, pp. 616–621.