Projective manifolds X with nef anticanonical bundles (i.e.
- K X ∙ C = det T X ∙ C ≥ 0 for all curves C ⊂ X ) can be regarded as an interpolation
between Fano manifolds (ample anticanonical bundle) and Calabi-Yau manifolds
resp. tori and symplectic manifolds (trivial canonical bundle). A differential-geometric
analogue are varieties with semi-positive Ricci curvature although this class is strictly
smaller -- to get the correct picture one has to consider sequences of metrics and make
the negative part smaller and smaller. However we will work completely in the context
of algebraic geometry.
Our aim is twofold: classification and, as a consequence, boundedness in case of
dimension 3. We shall not consider threefolds with trivial canonical bundles, the
eventual boundedness of Calabi-Yau threefolds still being unknown. Fano threefolds
have been classified a long time ago and threefolds with big and nef anticanonical
bundle are very much related with Q-Fano threefolds; therefore we will concentrate
here on projective threefolds X with - KX nef and K3X
= 0, but KX ≢ 0.
The essential problem is to distinguish the positive and flat directions in X.