Volume and Self-Intersection of Differences of Two Nef Classes

Dan Popovici (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

Let {α} and {β} be nef cohomology classes of bidegree (1, 1) on a compact n-dimensional Kähler manifold X such that the difference of intersection numbers {α} n − n {α} n−1. {β} is positive. We solve in a number of special but rather inclusive cases the quantitative part of Demailly's Transcendental Morse Inequalities Conjecture for this context predicting the lower bound {α} n − n {α} n−1. {β} for the volume of the difference class {α − β}. We completely solved the qualitative part in an earlier work. We also give general lower bounds for the volume of {α−β} and show that the self-intersection number {α−β} n is always bounded below by {α} n − n {α} n−1. {β}. We also describe and estimate the relative psef and nef thresholds of {α} with respect to {β} and relate them to the volume of {α − β}. Finally, broadening the scope beyond the Kähler realm, we propose a conjecture relating the balanced and the Gauduchon cones of ∂ ¯ ∂-manifolds which, if proved to hold, would imply the existence of a balanced metric on any ∂ ¯ ∂-manifold. Mathematics Subject Classification (2010): 32J27, 32U40, 32Q25, 32J25, 53C55.

Domaines: Mathématiques [math]

Dan Popovici. Volume and Self-Intersection of Differences of Two Nef Classes. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2016. ⟨hal-01976319⟩