CHAPTER 1
Introduction
1.1 Duality for elliptic fibrations
In this paper we are concerned with categories of sheaves on varieties
fibered by genus one curves. For an elliptic fibration on X, by which we
always mean a genus one fibration TT : X » B admitting a holomorphic
section a : B —» X, there is by now a well understood theory of the
Fourier-Mukai transform [Muk81, BBRP98, Bri98, BM02]. The
basic result is:
Theorem [BM02] Let X^_j^B be an elliptic fibration with smooth
a
total space. Then the integral transform (Fourier-Mukai transform)
FM : D\X) D\X)
F Rp2*(Lp\F g £*),
induced by the Poincare sheaf & X xBX, is an auto-equivalence of
the bounded derived category Db(X) of coherent sheaves on X.
An important feature of FM is that it transforms geometric objects
in an interesting way:
[Bundle data: vector bundles on X, semistable of
(^degree zero on the generic fiber of TT.
FM
{
Spectral data: sheaves on X with the numerics of]
a line bundle on a 'spectral' divisor C d , finite
over B.
This spectral construction was used to study general compactifica-
tions of heterotic string theory and their moduli, and especially the du-
ality with F-theory [FMW97, BJPS97, Don97, AD98, FMW98,
l
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