Communications in Analysis and Geometry

Volume 12 (2004)

Number 5

The Dual Kähler Cone of Compact Kähler Threefolds

Pages: 1131 – 1154

DOI: http://dx.doi.org/10.4310/CAG.2004.v12.n5.a7

Authors

Keiji Oguiso

Thomas Peternell

Abstract

The Kähler cone and its dual play an important role in the study of compact Kähler manifolds. Therefore it seems natural to ask whether one can read off the (dual) Kähler cone, whether the underlying manifold is projective or not. A classical theorem of Kodaira says that a compact Kähler manifold whose Kähler cone has an interior rational point, is projective. Indeed, a multiple of such a point defines a Hodge metric on the manifold. But not only the Kähler cone itself, also its dual in Hn-1,n-1(X) (1.7), where n is the dimension of the Kähler manifold X, is of interest. This is parallel to the projective theory, where both the ample cone and its dual cone NE, the Mori cone of curves, play an important role. One of the basic questions underlying this paper asks whether projectivity of Kähler manifolds can be expressed in terms of the dual Kähler cone. This question was first posed and treated for surfaces by Huybrechts [Hu99]; an geometric proof for surfaces was given in [OP00].

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