(see also Chern-Weil theory, parameterized homotopy theory)
An elliptic fibration is a bundle of elliptic curves, possibly including some singular fibers.
An elliptic surface is an elliptic fibration over an algebraic curve.
Write $SL_2(\mathbb{Z})$ for the special linear group in dimension 2 with integer coefficients and write $SL_2(\mathbb{Z}) \to PSL_2(\mathbb{Z})$ for the projection to the corresponding projective linear group. Regarding this as the Möbius group it comes with its natural action on the upper half plane $\mathfrak{h}$. The homotopy quotient $\mathcal{M}_{ell}(\mathbb{C}) = \mathfrak{h}//SL_2(\mathbb{Z})$ is the moduli stack of elliptic curves over the complex numbers.
Accordingly, to any $SL_2(\mathbb{Z})$-principal bundle $P \to B$ (necessarily flat since $SL_2(\mathbb{Z})$ is a discrete group, hence a “local system”) is associated a $\mathfrak{h}$-fiber bundle such that a section of it defines a non-singular elliptic fibration.
One may turn this around: Given an elliptic fibration $E \to B$, then away from the points $S\subset B$ over which the fiber is singular, it is given by an $SL_2(\mathbb{Z})$-local system together with a section of the associated upper-half plane bundle on $B-S$.
With due technical care, this data uniquely characterizes the elliptic fibration (e.g. Miranda 88, prop. VI.3.3).
The remaining singular fibers follow an ADE classification (Kodaira 64, Néron 64, Kodaira 66)
See at elliptic fibration of a K3-surface.
Specifically for complex surfaces:
Rick Miranda, The basic theory of elliptic surfaces, lecture notes 1988 (pdf)
Robert Friedman, John Morgan, Smooth Four-Manifolds and Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics (1994) (doi:10.1007/978-3-662-03028-8)
Fedor Bogomolov, Yuri Tschinkel, Monodromy of elliptic surfaces (pdf)
Takahiko Yoshida, Locally standard torus fibrations pdf
See also:
Specifically for elliptically fibered K3-surfaces:
The ADE classification of the possible singular fibers is due to
Kunihiko Kodaira, (1964). “On the structure of compact complex analytic surfaces. I”. Am. J. Math. 86: 751–798. doi:10.2307/2373157. Zbl 0137.17501.
Kunihiko Kodaira, (1966). “On the structure of compact complex analytic surfaces. II”. Am. J. Math. 88: 682–721. doi:10.2307/2373150. Zbl 0193.37701.
and
Last revised on February 7, 2021 at 14:44:55. See the history of this page for a list of all contributions to it.