Characterizing volume via cone duality

@article{Xiao2015CharacterizingVV,
  title={Characterizing volume via cone duality},
  author={Jian Xiao},
  journal={Mathematische Annalen},
  year={2015},
  volume={369},
  pages={1527-1555}
}
  • Jian Xiao
  • Published 23 February 2015
  • Mathematics
  • Mathematische Annalen
For divisors over smooth projective varieties, we show that the volume can be characterized by the duality between the pseudo-effective cone of divisors and the movable cone of curves. Inspired by this result, we define and study a natural intersection-theoretic volume functional for 1-cycles over compact Kähler manifolds. In particular, for numerical equivalence classes of curves over projective varieties, it is closely related to the mobility functional studied by Lehmann. 

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