Beyond the Kähler Cone

The moduli space of nonlinear σ-models on a Calabi–Yau manifold contains a complexification of the Kähler cone of the manifold. We describe a physically natural analytic continuation process which links the complexified Kähler cones of birationally equivalent Calabi–Yau manifolds. The enlarged moduli space includes a complexification of Kawamata’s “movable cone”. We formulate a natural conjecture about the action of the birational automorphism group on this cone. Many mathematicians were taken by surprise during 1984–85 when we found physicists knocking on our doors, asking whether we knew anything about Riemannian 6-manifolds with a metric whose holonomy lies in SU(3). Fortunately, Yau had solved the Calabi conjecture nearly 10 years earlier, so we were able to provide some answers: any smooth complex projective threefold with trivial canonical bundle admits a metric of this type. Also fortunately, these manifolds—now called Calabi–Yau threefolds—had been studied in some detail by algebraic geometers, in part due to the distinguished rôle they play in the classification theory of algebraic varieties. During the following year, the questions became more and more specific, focusing primarily on the physicists’ desire for examples X whose Euler number 1991 Mathematics Subject Classification. Primary 14J30; Secondary 14E07, 14J15, 32J81, 58D27, 81T40. Research partially supported by National Science Foundation Grant DMS-9103827, and by an American Mathematical Society Centennial Fellowship. Typeset by AMS-TEX 1