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Differentiability of volumes of divisors and a problem of Teissier. (English) Zbl 1162.14003
Let $$L$$ be a line bundle on an $$n$$-dimensional projective variety $$X$$. The volume of $$L$$ measures the rate of growth of the dimensions of the spaces of global sections $$H^0(X,kL)$$. It is defined by $\text{vol}(L)=\text{lim sup}_{k\to \infty} \frac {n!}{k^n}h^0(X,kL).$ It is known that $$\text{vol}(L)>0$$ if and only if $$H^0(X,kL)$$ defines a birational map for some $$k>0$$. In this case we say that $$L$$ is big. It is also the case that the volume of $$L$$ only depends on its numerical class $$[L]\in N^1(X)$$ and that it defines a continuous function on $$N^1(X)$$ such that $$\text{vol}^{1/n}$$ is concave and homogeneous of degree $$1$$. In the article under review, it is shown that the volume function is $$\mathcal C^1$$-differentiable on the big cone of $$N^1(X)$$ and for any $$\alpha, \gamma \in N^1(X)$$ such that $$\alpha$$ is big, we have $\frac d{dt}\Big| _{t=0} \text{vol}(\alpha +t \gamma )=n<\alpha ^{n-1}>\cdot \gamma .$ Here $$<\alpha ^{n-1}>\in N^1(X)^*$$ denotes the positive intersection product of the big class $$\alpha$$. When $$D$$ is a prime divisor and $$\gamma$$ is the numerical class of $$D$$, the authors also show that $$<\alpha ^{n-1}>\cdot \gamma$$ is given by the restricted volume $$\text{vol }_{X|D}(L)=\text{lim sup}_{k\to \infty} \frac {(n-1)!}{k^{n-1}}h^0(X|V,kL)$$ where $$h^0(X|V,kL)$$ denotes the dimension of the image of the restriction homomorphism $$H^0(X,kL)\to H^0(V,kL|_V)$$. The authors then show that the function $$\alpha \to (\alpha ^n)^{1/n}$$ is strictly concave on the big and nef cone of $$N^1(X)$$, they give a result characterizing the equality case of the Khovanskii-Teissier inequalities and they prove an algebro-geometric version of the Diskant inequality in convex geometry.

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves
##### Keywords:
volumes of divisors
Full Text:
##### References:
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