Nef Reduction and Anticanonical Bundles

Abstract

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 - K_X nef and K³_X = 0, but K_X ≢ 0. The essential problem is to distinguish the positive and flat directions in X.