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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.

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
Full Text: DOI
[1] Björk, J.-E., Analytic \(\mathcal{D}\)-modules and applications, (1993), Kluwer Academic Publisher: Kluwer Academic Publisher, Dordrecht
[2] Dimca, A., Sheaves in topology, (2004), Springer-Verlag: Springer-Verlag, Berlin, New York · Zbl 1043.14003
[3] Dimca, A.; Maisonobe, Ph.; Saito, Morihiko; Torrelli, T., Multiplier ideals, \(V\)-filtrations and transversal sections, Math. Ann., 336, 4, 901-924, (2006) · Zbl 1107.14003
[4] Dutta, Yajnaseni, Vanishing for Hodge ideals on toric varieties, (2018)
[5] Fulton, W., Introduction to toric varieties, 131, (1993), Princeton University Press: Princeton University Press, Princeton, NJ
[6] Greb, Daniel; Kebekus, Stefan; Kovács, Sándor J.; Peternell, Thomas, Differential forms on log canonical spaces, Publ. Math. Inst. Hautes Études Sci., 114, 87-169, (2011) · Zbl 1258.14021
[7] Hotta, R.; Takeuchi, K.; Tanisaki, T., \(D\)-modules, perverse sheaves, and representation theory, 236, (2008), Birkhäuser: Birkhäuser, Boston, Basel, Berlin · Zbl 1136.14009
[8] Lazarsfeld, R., Positivity in algebraic geometry. II, 49, (2004), Springer-Verlag: Springer-Verlag, Berlin · Zbl 1093.14500
[9] Mustaţă, Mircea; Popa, Mihnea, Hodge ideals, (2016)
[10] Mustaţă, Mircea; Popa, Mihnea, Hodge ideals for \(\mathbf{Q}\)-divisors, \(V\)-filtration, and minimal exponent, (2018) · Zbl 1408.14074
[11] Mustaţă, Mircea; Popa, Mihnea, Restriction, subadditivity, and semicontinuity theorems for Hodge ideals, Internat. Math. Res. Notices, 11, 3587-3605, (2018) · Zbl 1408.14074
[12] Popa, Mihnea, Proceedings of the ICM (Rio de Janeiro, 2018), Vol. 2, \(\mathscr{D}\)-modules in birational geometry, 781-806, (2019), World Scientific Publishing Co., Inc.: World Scientific Publishing Co., Inc., River Edge, NJ
[13] Saito, Morihiko, Algebraic cycles and motives. Vol. 2, 344, Direct image of logarithmic complexes and infinitesimal invariants of cycles, 304-318, (2007), Cambridge Univ. Press: Cambridge Univ. Press, Cambridge · Zbl 1125.14007
[14] Saito, Morihiko, On the Hodge filtration of Hodge modules, Moscow Math. J., 9, 1, 161-191, (2009) · Zbl 1196.14015
[15] Saito, Morihiko, Hodge ideals and microlocal \(V\)-filtration, (2016)
[16] Saito, Morihiko, Hodge theory and \(L^2\)-analysis, 39, A young person’s guide to mixed Hodge modules, 517-553, (2017), Int. Press: Int. Press, Somerville, MA · Zbl 1379.14001
[17] Saito, Morihiko, Modules de Hodge polarisables, Publ. RIMS, Kyoto Univ., 24, 6, 849-995, (1988) · Zbl 0691.14007
[18] Saito, Morihiko, Mixed Hodge modules, Publ. RIMS, Kyoto Univ., 26, 2, 221-333, (1990) · Zbl 0727.14004
[19] Zhang, Mingyi, Hodge filtration for \(\mathbf{Q}\)-divisors with quasi-homogeneous isolated singularities, (2018)
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