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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
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