×

zbMATH — the first resource for mathematics

\(E_1\)-degeneration of the irregular Hodge filtration. (English) Zbl 06762464
Summary: For a regular function \(f\) on a smooth complex quasi-projective variety, J.-D. Yu introduced in [Manuscr. Math. 144, No. 1–2, 99–133 (2014; Zbl 1291.14040)] a filtration (the irregular Hodge filtration) on the de Rham complex with twisted differential \(\mathrm{d}+\mathrm{d}f\), extending a definition of Deligne in the case of curves. In this article, we show the degeneration at \(E_1\) of the spectral sequence attached to the irregular Hodge filtration, by using the method of C. Sabbah [Adv. Stud. Pure Math. 59, 289–347 (2010; Zbl 1264.14011)]. We also make explicit the relation with a complex introduced by M. Kontsevich and give details on his proof of the corresponding \(E_1\)-degeneration, by reduction to characteristic \(p\), when the pole divisor of the function is reduced with normal crossings. In Appendix E, M. Saito gives a different proof of the \(E_1\)-degeneration.

MSC:
32C38 Sheaves of differential operators and their modules, \(D\)-modules
14D07 Variation of Hodge structures (algebro-geometric aspects)
18G40 Spectral sequences, hypercohomology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. Adolphson and S. Sperber, Exponential sums and Newton polyhedra, cohomology and estimates, Ann. of Math. (2) 130 (1989), 367-406. · Zbl 0723.14017
[2] A. A. Beilinson, How to glue perverse sheaves, K-theory, arithmetic and geometry (Moscow 1984-1986), Lecture Notes in Math. 1289, Springer, Berlin (1987), 42-51. · Zbl 0651.14009
[3] P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Math. 163, Springer, Berlin 1970.
[4] P. Deligne, Théorie de Hodge II, Publ. Math. Inst. Hautes Études Sci. 40 (1971), 5-57. · Zbl 0219.14007
[5] P. Deligne, Le formalisme des cycles évanescents (exposés 13 et 14), Seminaire de géométrie algébrique du Bois-Marie 1967-1969. Groupes de monodromie en géométrie algébrique (SGA 7 II), Lecture Notes in Math. 340, Springer, Berlin (1973), 82-173.
[6] P. Deligne, Théorie de Hodge irrégulière (mars 1984 & août 2006), Singularités irrégulières, correspondance et documents, Doc. Math. (Paris) 5, Société Mathématique de France, Paris (2007), 109-114, 115-128.
[7] P. Deligne and L. Illusie, Relèvements modulo p^2 et décomposition du complexe de de Rham, Invent. Math. 89 (1987), 247-270. · Zbl 0632.14017
[8] H. Esnault and E. Viehweg, Lectures on vanishing theorems, DMV Semin. 20, Birkhäuser, Basel 1992.
[9] H. Fan, Schrödinger equations, deformation theory and tt^*-geometry, preprint (2011), .
[10] M. Hien, Periods for flat algebraic connections, Invent. Math. 178 (2009), no. 1, 1-22. · Zbl 1190.14019
[11] L. Katzarkov, M. Kontsevich and T. Pantev, Bogomolov-Tian-Todorov theorems for Landau-Ginzburg models, preprint (2014), . · Zbl 1361.35172
[12] M. Kontsevich, Holonomic \mathcalD-modules and positive characteristic, Jpn. J. Math. 4 (2009), 1-25. · Zbl 1215.14014
[13] M. Kontsevich, Letters to L. Katzarkov and T. Pantev, (2012).
[14] M. Kontsevich, Letters to H. Esnault and J.-D. Yu, (2012).
[15] G. Laumon, Sur la catégorie dérivée des \mathcalD-modules filtrés, Algebraic geometry (Tokyo/Kyoto 1982), Lecture Notes in Math. 1016, Springer, Berlin (1983), 151-237. · Zbl 0551.14006
[16] H. Majima, Asymptotic analysis for integrable connections with irregular singular points, Lecture Notes in Math. 1075, Springer, Berlin 1984. · Zbl 0546.58003
[17] T. Mochizuki, Asymptotic behaviour of tame harmonic bundles and an application to pure twistor D-modules, Mem. Amer. Math. Soc. 185 (2007), 869-870.
[18] T. Mochizuki, Wild harmonic bundles and wild pure twistor D-modules, Astérisque 340, Société Mathématique de France, Paris 2011.
[19] A. Ogus and V. Vologodsky, Nonabelian Hodge theory in characteristic p, Publ. Math. Inst. Hautes Études Sci. 106 (2007), 1-138. · Zbl 1140.14007
[20] C. Sabbah, \mathcalD-modules et cycles évanescents (d’après B. Malgrange et M. Kashiwara), Géométrie algébrique et applications. Vol. III (La Rábida 1984), Hermann, Paris (1987), 53-98. · Zbl 0623.32013
[21] C. Sabbah, On the comparison theorem for elementary irregular \mathcalD-modules, Nagoya Math. J. 141 (1996), 107-124. · Zbl 0858.32013
[22] C. Sabbah, Monodromy at infinity and Fourier transform, Publ. Res. Inst. Math. Sci. 33 (1997), no. 4, 643-685. · Zbl 0920.14003
[23] C. Sabbah, On a twisted de Rham complex, Tôhoku Math. J. 51 (1999), 125-140. · Zbl 0947.14007
[24] C. Sabbah, Polarizable twistor \mathcalD-modules, Astérisque 300, Société Mathématique de France, Paris 2005.
[25] C. Sabbah, Monodromy at infinity and Fourier transform II, Publ. Res. Inst. Math. Sci. 42 (2006), 803-835. · Zbl 1259.14008
[26] C. Sabbah, Fourier-Laplace transform of a variation of polarized complex Hodge structure, II, New developments in algebraic geometry, integrable systems and mirror symmetry (Kyoto 2008), Adv. Stud. Pure Math. 59, Mathematical Society of Japan, Tokyo (2010), 289-347. · Zbl 1264.14011
[27] M. Saito, Hodge filtrations on Gauss-Manin systems. II, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 2, 37-40. · Zbl 0549.32017
[28] M. Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), 849-995. · Zbl 0691.14007
[29] M. Saito, Induced \mathcalD-modules and differential complexes, Bull. Soc. Math. France 117 (1989), 361-387. · Zbl 0705.32005
[30] M. Saito, Extension of mixed Hodge modules, Compos. Math. 74 (1990), no. 2, 209-234. · Zbl 0726.14007
[31] M. Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), 221-333. · Zbl 0727.14004
[32] J. H. M. Steenbrink, Limits of Hodge structures, Invent. Math. 31 (1976), 229-257. · Zbl 0303.14002
[33] J. H. M. Steenbrink, Mixed Hodge structure on the vanishing cohomology, Real and complex singularities (Oslo 1976), Sijthoff and Noordhoff, Alphen aan den Rijn (1977), 525-563. · Zbl 0373.14007
[34] J. H. M. Steenbrink and S. Zucker, Variation of mixed Hodge structure I, Invent. Math. 80 (1985), 489-542. · Zbl 0626.14007
[35] J.-D. Yu, Irregular Hodge filtration on twisted de Rham cohomology, Manuscripta Math. 144 (2014), no. 1-2, 99-133. · Zbl 1291.14040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.