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Hodge ideals for $$\mathbf{Q}$$-divisors: birational approach. (Idéaux de Hodge pour des $$\mathbf{Q}$$-diviseurs: approche birationnelle.) (English. French summary) Zbl 1427.14045
Let $$X$$ be a nonsingular complex variety and let $$D$$ be a reduced divisor on $$X$$. Then the Hodge ideals $$I_k(D)$$, $$k \geq 0$$, are defined in terms of the Hodge filtration on the $$\mathcal D_X$$-module $$\mathcal O_X(\star D)$$ of functions with poles of any order along $$D$$ so that $$F_k\mathcal O_X(\star D) = I_k(D)\otimes \mathcal O_X ((k + 1)D)$$ for all $$k \geq 0$$, where $$F_\bullet (\mathcal O_X(\star D))$$ is a Hodge filtration (see [M. Mustaţă and M. Popa, “Hodge ideals”, arXiv:1605.08088]).
In the paper under review, the authors describe a similar construction for effective $$\mathbb Q$$-divisors on $$X$$ using tools of the theory of log resolutions. Then they prove local nontriviality criteria, global and local vanishing theorems, semicontinuity theorem and other analogues of standard results from the theory of multiplier ideals. In addition, they also discuss a series of applications (see [M. Mustaţă and M. Popa, Int. Math. Res. Not. 2018, No. 11, 3587–3605 (2018; Zbl 1408.14074)]), etc.

##### MSC:
 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14F18 Multiplier ideals 14C20 Divisors, linear systems, invertible sheaves 14J17 Singularities of surfaces or higher-dimensional varieties 32S25 Complex surface and hypersurface singularities 14F17 Vanishing theorems in algebraic geometry
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