# 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} }

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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## References

SHOWING 1-10 OF 36 REFERENCES

### Zariski decomposition of curves on algebraic varieties

- Mathematics
- 2015

We introduce a Zariski decomposition for curve classes and use it to develop the theory of the volume function for curves defined by the second named author. For toric varieties and for hyperk\"ahler…

### Geometric characterizations of big cycles

- Mathematics
- 2013

A numerical equivalence class of k-cycles is said to be big if it lies in the interior of the closed cone generated by effective classes. We develop several geometric criteria that distinguish big…

### The pseudo-effective cone of a compact K\

- Mathematics
- 2004

We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a…

### Pseudoeffective and nef classes on abelian varieties

- MathematicsCompositio Mathematica
- 2011

Abstract We study the cones of pseudoeffective and nef cycles of higher codimension on the self product of an elliptic curve with complex multiplication, and on the product of a very general abelian…

### Zariski decompositions of numerical cycle classes

- Mathematics
- 2013

We construct a Zariski decomposition for cycle classes of arbitrary codimension. This decomposition is an analogue of well-known constructions for divisors. Examples illustrate how Zariski…

### Holomorphic Morse Inequalities and the Green-Griffiths-Lang Conjecture

- Mathematics
- 2010

The goal of this work is to study the existence and properties of non constant entire curves f drawn in a complex irreducible n-dimensional variety X, and more specifically to show that they must…

### ON THE VOLUME OF A LINE BUNDLE

- Mathematics
- 2002

Using the Calabi–Yau technique to solve Monge-Ampere equations, we translate a result of T. Fujita on approximate Zariski decompositions into an analytic setting and combine this to the holomorphic…

### On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I*

- Mathematics
- 1978

Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent the…