Teissier's problem on inequalities of nef divisors

Abstract

Teissier has proven remarkable inequalities [Formula: see text] for intersection numbers si = (ℒi ⋅ ℳd-i) of a pair of nef line bundles ℒ, ℳ on a d-dimensional complete algebraic variety over a field. He asks if two nef and big line bundles are numerically proportional if the inequalities are all equalities. In this paper, we show that this is true in the most general possible situation, for nef and big line bundles on a proper irreducible scheme over an arbitrary field k. Boucksom, Favre and Jonsson have recently established this result on a complete variety X over an algebraically closed field of characteristic zero. Their proof involves an ingenious extension of the intersection theory on a variety to its Zariski Riemann Manifold. This extension requires the existence of a direct system of nonsingular varieties dominating X. We make use of a simpler intersection theory which does not require resolution of singularities, and extend volume to an arbitrary field and prove its continuous differentiability, extending results of Boucksom, Favre and Jonsson, and of Lazarsfeld and Mustaţă. A goal in this paper is to provide a manuscript which will be accessible to many readers. As such, subtle topological arguments which are required to give a complete proof in [S. Bouksom et al., J. Algebraic Geometry18 (2009) 279–308] have been written out in this manuscript, in the context of our intersection theory, and over arbitrary varieties.