.4. (see [Xia15,Definition 1.1]) Let X be a projective variety of dimension n and let α ∈ Eff 1 (X) be a pseudo-effective curve class. Then the volume of α is defined to be vol(α) = inf A big and nef divisor class A · α vol(A) 1/n n n−1.
We say that a big and nef divisor class A computes vol(α) if this infimum is achieved by A. When α is a curve class that is not pseudo-effective, we set vol(α) = 0.
The function vol is a polar transformation of the volume function for ample divisors. In our setting, the polar transformation plays the role of the Legendre-Fenchel transform of classical convex analysis, linking the differentiability of a function to the strict convexity of its transform. From this viewpoint, Definition 1.1 is important precisely because it captures the log concavity of vol.
As an important special case, the positive part of a curve class has the same volume as the original class, showing the similarity with the σ-decomposition. Furthermore, just in Zariski's classical work, the "projection" onto the positive part elucidates the intersection-theoretic nature of the volume.
Theorem 1.6. Let X be a projective variety of dimension n and let α ∈ Eff 1 (X) • be a big curve class. Suppose that α = B n−1 + γ is the Zariski decomposition of α. Then vol(α) = vol(B n−1 ) = B n and B is the unique big and nef divisor class with this property satisfying B n−1 α.
Example 1.7. An important feature of Zariski decompositions and vol for curves is that they can be calculated via intersection theory directly on X once one has identified the nef cone of divisors. (In contrast, the analogous divisor constructions may require passing to birational models of X to admit an interpretation via intersection theory.) This is illustrated by Example 5.5 where we calculate the Zariski decomposition of any curve class on the projective bundle over P 1 defined by O ⊕ O ⊕ O(−1).
1.3. Formal Zariski decompositions. The Zariski decomposition for curves can be deduced from a general theory of duality for log concave homogeneous functions defined on cones. We define a "formal" Zariski decomposition capturing the failure of strict log concavity of a certain class of homogeneous functions on finite-dimensional cones.
Let C be a full dimensional closed proper convex cone in a finite dimensional vector space. For any s > 1, let HConc s (C) denote the collection of functions f : C → R that are upper-semicontinuous, homogeneous of weight s > 1, strictly positive on the interior of C, and which are s-concave in the sense that f (v) 1/s + f (x) 1/s ≤ f (x + v) 1/s for any v, x ∈ C. In this context, the correct analogue of the Legendre-Fenchel transform is the (concave homogeneous) polar transform. For any f ∈ HConc s (C), the polar Hf is an element of HConc s/s−1 (C * ) for the dual cone C * defined as We define what it means for f ∈ HConc s (C) to have a Zariski decomposition structure and show that it follows from a differentiability condition for Hf, and vice versa (see Section 4). Just as in the classical definition of Zariski, one can view this structure as a decomposition of the elements of C • into "positive parts" retaining the value of f and "negative parts" along which the strict log concavity of f fails.
Example 1.8. Let q be a bilinear form on a vector space V of signature (1, dim V − 1) and set f (v) = q(v, v). Suppose C is a closed full-dimensional convex cone on which f is non-negative. Identifying V with V * under q, we see that C ⊂ C * and that Hf | C = f by the Hodge inequality. Then Hf on the entire cone C * is controlled by a Zariski decomposition with positive parts lying in C.
This is of course the familiar picture for surfaces, where f is the self-intersection on the nef cone and Hf is the volume on the pseudo-effective cone. Thus we see that the conclusion of Example 1.3 -that vol and vol coincide on surfaces -is a direct consequence of the Hodge Index Theorem for surfaces. Furthermore, we obtain a theoretical perspective motivating the linear algebra calculations of [Zar62].
Many of the basic geometric inequalities in algebraic geometry -and hence for polytopes or convex bodies via toric varieties (as in [Tei82] and [Kho89] and the references therein) -can be understood using this abstract framework. A posteriori this theory motivates many well-known theorems about the volume of divisors (which can itself be interpreted as a polar transform). In particular, the σdecomposition for divisor classes can be also interpreted by our general theory. See Remark 5.25 for more details.
1.4. Positivity of curves. The volume function for curves shares many of the important properties of the volume function for divisors. This is no accident -as explained above, polar duality behaves compatibly with many topological properties and with geometric inequalities. Clearly the volume function is homogeneous and it is not hard to show that it is positive precisely on the big cone of curves. Perhaps the most important property is the following description of the derivative, which mirrors the results of [BFJ09] and [LM09] for divisors. Theorem 1.9. Let X be a projective variety of dimension n. Then the function vol is C 1 on the big cone of curves. More precisely, let α be a big curve class on X and write α = B n−1 + γ for its Zariski decomposition. For any curve class β, we have d dt t=0 vol(α + tβ) = n n − 1 B · β.
Another key property of the σ-decomposition for divisors is that the negative part is effective. While the negative part of the Zariski decomposition for curves need not be effective, the correct analogue is given by the following proposition.
Proposition 1.10. Let X be a projective variety of dimension n. Let α be a big curve class and write α = B n−1 + γ for its Zariski decomposition. There is a proper subscheme i : V X and a pseudo-effective class γ ′ ∈ N 1 (V ) such that i * γ ′ = γ.
By analogy with the algebraic Morse inequality for nef divisors, we prove a Morse-type inequality for curves.
Theorem 1.11. Let X be a smooth projective variety of dimension n. Let α be a big curve class and let β be a nef curve class. Write α = B n−1 + γ for the Zariski decomposition of α. If vol(α) − nB · β > 0 then α − β is big.
1.5. Examples. The Zariski decomposition is particularly striking for varieties with a rich geometric structure. We discuss several examples: toric varieties, Mori dream spaces and hyperkähler manifolds.
The complete intersection cone CI 1 (X) is defined to be the closure of the set of classes of the form A n−1 for an ample divisor A on X. Note that the positive part of the Zariski decomposition takes values in CI 1 (X). We should emphasize that CI 1 (X) need not be convex -the appendix gives an explicit example. 1.5.1. Toric varieties. Let X be a simplicial projective toric variety of dimension n defined by a fan Σ. Suppose that the curve class α lies in the interior of the movable cone of curves, or equivalently, α is defined by a positive Minkowski weight on the rays of Σ. A classical theorem of Minkowski attaches to such a weight a polytope P α whose facet normals are the rays of Σ and whose facet volumes are determined by the weights.
In this setting, the volume is calculated by an mixed volume problem: fixing P α, amongst all polytopes whose normal fan refines Σ there is a unique Q (up to homothety) minimizing the mixed volume calculation Then the volume is n! times this minimum value, and the positive part of α is proportional to the (n − 1)-product of the big and nef divisor defined by Q. Note that if we let Q vary over all polytopes then the Brunn-Minkowski inequality shows that the minimum is given by Q = cP α, but the normal fan condition on Q yields a new version of this classical problem.
We give a procedure for computing the volume of any big curve class α using Zariski decompositions. From the viewpoint of convex analysis, the compatibility with the Zariski decomposition corresponds to the fact that the solution of an mixed volume problem should be given by a condition on the derivative.
The procedure is as follows. Note that every big and nef divisor on X is semi-ample (that is, the pullback of an ample divisor on a toric birational model). Thus, the Zariski decomposition for curves is characterized by the existence of a birational toric morphism π : X → X ′ such that: • the class π * α ∈ N 1 (X ′ ) coincides with A n−1 for some