nLab elliptic fibration

Idea

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.

Properties

Classification by local systems with modular group coefficients

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).

Classification of the singular fibers

The remaining singular fibers follow an ADE classification (Kodaira 64, Néron 64, Kodaira 66)

Examples

Elliptic fibration of K3-surfaces

See at elliptic fibration of a K3-surface.

Related entries

• F-theory

  • F/M-theory on elliptically fibered Calabi-Yau 4-folds

• K3

References

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, arXiv:math/0002168]

• Takahiko Yoshida, Locally standard torus fibrations pdf

See also:

• Wikipedia, Elliptic surface

Specifically for elliptically fibered K3-surfaces:

• Viacheslav Nikulin, Elliptic fibrations on K3 surfaces (arXiv:1010.3904)

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

• André Néron, (1964). “Modeles minimaux des variétés abeliennes sur les corps locaux et globaux”. Publications Mathématiques de l’IHÉS (in French) 21: 5–128.