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Local volumes of cartier divisors over normal algebraic varieties. (Volumes locaux de diviseurs de cartier sur des variétés algébriques normales.) (English. French summary) Zbl 1297.14015

In the article under review, the author introduces a notion of local volume for Cartier divisors.
Let \(X\) be a normal quasiprojective variety of dimension \(n\geq 2\) over \(\mathbb{C}\), and let \(x\) be a point on \(X\). Fix a projective birational morphism \(\pi: X'\rightarrow X\), and let \(D\) be a Cartier divisor on \(X'\). The local volume of \(D\) at \(x\) is defined as \[ \mathrm{vol}_x (D) := \overline{\lim}_{m\rightarrow +\infty} \frac{n ! \cdot h_x ^1 (m D)}{m^n} , \] where \(h_x ^1 (m D) := \dim H_{{x}} ^1 (X, \pi_* \mathcal{O}_{X'} (D))\).
The author proved several important properties of the local volume. As a main theorem, the author proved that \(\mathrm{vol}_x\) is a well-defined \(n\)-homogeneous, and locally Lipschitz continuous on \(N^1 (X' /X)_{\mathbb{R}}\). The author also studies the vanishing and convexity for \(\mathrm{vol}_x ^{\frac{1}{n}}\).
In the last part of the article, the author compare the local volume with another notion of volume defined in [S. Boucksom, T. De Fernex and C. Favre, Duke Math. J. 161, No. 8, 1455–1520 (2012; Zbl 1251.14026)]. Let \((\widetilde{X}, E)\rightarrow (X, x)\) be a log-resolution of a normal isolated singularity. We set \(\mathrm{vol} (X, x) := \mathrm{vol}_x (K_{\widetilde{X} +E})\) and set \(\mathrm{vol}_{BdFF} (X,x)\) be the local volume defined in [loc. cit.]. In general, we have \(\mathrm{vol} (X, x) \leq \mathrm{vol}_{BdFF} (X,x)\). The author proved that these two definitions coincide in the numerically Gorenstein case.
Reviewer: Junyan Cao (Paris)

MSC:

14E05 Rational and birational maps
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14B05 Singularities in algebraic geometry
14B15 Local cohomology and algebraic geometry
32S05 Local complex singularities

Citations:

Zbl 1251.14026
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References:

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