Non-commutative Hodge structures
Annales de l'Institut Fourier, Volume 61 (2011) no. 7, p. 2681-2717

This article gives a survey of recent results on a generalization of the notion of a Hodge structure. The main example is related to the Fourier-Laplace transform of a variation of polarizable Hodge structure on the punctured affine line, like the Gauss-Manin systems of a proper or tame algebraic function on a smooth quasi-projective variety. Variations of non-commutative Hodge structures often occur on the tangent bundle of Frobenius manifolds, giving rise to a tt* geometry.

Nous donnons un panorama des résultats récents concernant une généralisation de la notion de structure de Hodge. L’exemple principal est celui produit par la transformation de Fourier-Laplace d’une variation de structure de Hodge polarisable sur la droite affine épointée, comme les systèmes de Gauss-Manin de fonctions algébriques propres ou modérées sur une variété quasi-projective lisse complexe. Le fibré tangent d’une variété de Frobenius peut souvent être muni d’une variation de structures de Hodge non-commutatives polarisables, d’où l’on déduit une géométrie spéciale du type tt*.

DOI : https://doi.org/10.5802/aif.2790
Classification:  14D07,  34M40
Keywords: Non-commutative Hodge structure, Fourier-Laplace transformation, Brieskorn lattice
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     author = {Sabbah, Claude},
     title = {Non-commutative Hodge structures},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {7},
     year = {2011},
     pages = {2681-2717},
     doi = {10.5802/aif.2790},
     mrnumber = {3112504},
     zbl = {pre06193023},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_7_2681_0}
}
Sabbah, Claude. Non-commutative Hodge structures. Annales de l'Institut Fourier, Volume 61 (2011) no. 7, pp. 2681-2717. doi : 10.5802/aif.2790. http://www.numdam.org/item/AIF_2011__61_7_2681_0/

[1] Balser, W.; Jurkat, W. B.; Lutz, D. A. On the reduction of connection problems for differential equations with an irregular singular point to ones with only regular singularities. I, SIAM J. Math. Anal., Tome 12 (1981) no. 5, pp. 691-721 | Article | MR 625827 | Zbl 0468.34024

[2] Borel, A. Chap. VI-IX, Algebraic 𝒟 -modules, Academic Press, Boston (Perspectives in Mathematics) Tome 2 (1987), pp. 207-352 | MR 882000 | Zbl 0642.32001

[3] Cattani, Eduardo; Kaplan, Aroldo; Schmid, Wilfried Degeneration of Hodge structures, Ann. of Math. (2), Tome 123 (1986) no. 3, pp. 457-535 | Article | MR 840721 | Zbl 0617.14005

[4] Cecotti, Sergio; Fendley, Paul; Intriligator, Ken; Vafa, Cumrun A new supersymmetric index, Nuclear Phys. B, Tome 386 (1992) no. 2, pp. 405-452 | Article | MR 1193656

[5] Cecotti, Sergio; Vafa, Cumrun Topological–anti-topological fusion, Nuclear Phys. B, Tome 367 (1991) no. 2, pp. 359-461 | Article | MR 1139739 | Zbl 1136.81403

[6] Cecotti, Sergio; Vafa, Cumrun On classification of N=2 supersymmetric theories, Comm. Math. Phys., Tome 158 (1993) no. 3, pp. 569-644 http://projecteuclid.org/getRecord?id=euclid.cmp/1104254363 | MR 1255428 | Zbl 0787.58049

[7] Deligne, P. Lettre à B. Malgrange du 19 avril 1978, Singularités irrégulières, Correspondance et documents, Société Mathématique de France, Paris (Documents mathématiques) Tome 5 (2007), p. 25-26 | MR 2387754 | Zbl 1130.14001

[8] Deligne, P. Théorie de Hodge irrégulière (mars 1984 et août 2006), Singularités irrégulières, Correspondance et documents, Société Mathématique de France, Paris (Documents mathématiques) Tome 5 (2007), p. 109-114 & 115-128 | MR 2387754

[9] Hertling, Claus Frobenius manifolds and moduli spaces for singularities, Cambridge University Press, Cambridge, Cambridge Tracts in Mathematics, Tome 151 (2002) | Article | MR 1924259 | Zbl 1023.14018

[10] Hertling, Claus tt * geometry, Frobenius manifolds, their connections, and the construction for singularities, J. Reine Angew. Math., Tome 555 (2003), pp. 77-161 | Article | MR 1956595 | Zbl 1040.53095

[11] Hertling, Claus tt * geometry and mixed Hodge structures, Singularity theory and its applications, Math. Soc. Japan, Tokyo (Adv. Stud. Pure Math.) Tome 43 (2006), pp. 73-84 | MR 2313409 | Zbl 1127.14300

[12] Hertling, Claus; Sabbah, C. Examples of non-commutative Hodge structures, Journal de l’Institut mathématique de Jussieu, Tome 10 (2011) no. 3, pp. 635-674 | MR 2806464 | Zbl 1246.14019

[13] Hertling, Claus; Sevenheck, Christian Nilpotent orbits of a generalization of Hodge structures, J. Reine Angew. Math., Tome 609 (2007), pp. 23-80 | Article | MR 2350780 | Zbl 1136.32011

[14] Hertling, Claus; Sevenheck, Christian Curvature of classifying spaces for Brieskorn lattices, J. Geom. Phys., Tome 58 (2008) no. 11, pp. 1591-1606 | Article | MR 2463812 | Zbl 1160.14005

[15] Hertling, Claus; Sevenheck, Christian Twistor structures, tt * -geometry and singularity theory, From Hodge theory to integrability and TQFT tt*-geometry, Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 78 (2008), pp. 49-73 | MR 2483748 | Zbl 1151.14304

[16] Hertling, Claus; Sevenheck, Christian Limits of families of Brieskorn lattices and compactified classifying spaces, Adv. Math., Tome 223 (2010) no. 4, pp. 1155-1224 | Article | MR 2581368 | Zbl 1190.14011

[17] Iritani, Hiroshi An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math., Tome 222 (2009) no. 3, pp. 1016-1079 | Article | MR 2553377 | Zbl 1190.14054

[18] Iritani, Hiroshi tt*-geometry in quantum cohomology (2009) (arXiv: 0906.1307) | MR 2553377

[19] Kaledin, D. Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie, Pure Appl. Math. Q., Tome 4 (2008) no. 3, Special issue in honor of F. Bogomolov, Part 2, pp. 785-875 | MR 2435845 | Zbl 1189.14013

[20] Kaledin, D. Motivic structures in non-commutative geometry (2010) (arXiv: 1003.3210; to appear in Proc. ICM) | MR 2827805 | Zbl 1245.14005

[21] Kashiwara, Masaki The asymptotic behavior of a variation of polarized Hodge structure, Publ. Res. Inst. Math. Sci., Tome 21 (1985) no. 4, pp. 853-875 | Article | MR 817170 | Zbl 0594.14012

[22] Kashiwara, Masaki D -modules and microlocal calculus, American Mathematical Society, Providence, RI, Translations of Mathematical Monographs, Tome 217 (2003) | MR 1943036 | Zbl 1017.32012

[23] Katz, N. Exponential sums and differential equations, Princeton University Press, Princeton, NJ, Ann. of Math. studies, Tome 124 (1990) | MR 1081536 | Zbl 0731.14008

[24] Katz, N. Rigid local systems, Princeton University Press, Princeton, NJ, Ann. of Math. studies, Tome 139 (1996) | MR 1366651 | Zbl 0864.14013

[25] Katz, Nicholas M.; Laumon, Gérard Transformation de Fourier et majoration de sommes exponentielles, Inst. Hautes Études Sci. Publ. Math. (1985) no. 62, pp. 361-418 | Numdam | MR 823177 | Zbl 0603.14015

[26] Katzarkov, L.; Kontsevich, M.; Pantev, T. Hodge theoretic aspects of mirror symmetry, From Hodge theory to integrability and TQFT tt*-geometry, Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 78 (2008), pp. 87-174 | MR 2483750 | Zbl 1206.14009

[27] Kontsevich, M.; Soibelman, Y. Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants (2010) (arXiv: 1006.2706) | MR 2851153 | Zbl 1248.14060

[28] Malgrange, Bernard Équations différentielles à coefficients polynomiaux, Birkhäuser Boston Inc., Boston, MA, Progress in Mathematics, Tome 96 (1991) | MR 1117227 | Zbl 0764.32001

[29] Malgrange, Bernard Connexions méromorphes. II. Le réseau canonique, Invent. Math., Tome 124 (1996) no. 1-3, pp. 367-387 | Article | MR 1369422 | Zbl 0849.32003

[30] Mochizuki, Takuro Asymptotic behaviour of tame harmonic bundles and an application to pure twistor D-modules, Mem. Amer. Math. Soc., Tome 185 (2007) no. 869-870 | Zbl pre05119702

[31] Mochizuki, Takuro Holonomic 𝒟 -modules with Betti structure (2010) (arXiv: 1001.2336) | MR 2605752

[32] Mochizuki, Takuro Asymptotic behaviour of variation of pure polarized TERP structures, Publications of the Research Institute for Mathematical Sciences , Kyoto Univ., Tome 47 (2011) no. 2, pp. 419-534 (arXiv: 0811.1384) | MR 2849639 | Zbl 1231.32013

[33] Mochizuki, Takuro Wild harmonic bundles and wild pure twistor D-modules, Astérisque, Société Mathématique de France, Paris (2011) (to appear; arXiv: 0803.1344) | MR 2919903 | Zbl 1245.32001

[34] Peters, Chris A. M.; Steenbrink, Joseph H. M. Mixed Hodge structures, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Tome 52 (2008) | MR 2393625 | Zbl 1138.14002

[35] Sabbah, Claude Hypergeometric period for a tame polynomial, C. R. Acad. Sci. Paris Sér. I Math., Tome 328 (1999) no. 7, pp. 603-608 | Article | MR 1679978 | Zbl 0967.32028

[36] Sabbah, Claude Déformations isomonodromiques et variétés de Frobenius, EDP Sciences, Les Ulis, Savoirs Actuels (Les Ulis). [Current Scholarship (Les Ulis)] (2002) (Mathématiques (Les Ulis). [Mathematics (Les Ulis)]) | MR 1933784 | Zbl 1101.14001

[37] Sabbah, Claude Polarizable twistor 𝒟-modules, Astérisque (2005) no. 300, pp. vi+208 | MR 2156523 | Zbl 1085.32014

[38] Sabbah, Claude Hypergeometric periods for a tame polynomial, Port. Math. (N.S.), Tome 63 (2006) no. 2, pp. 173-226 | MR 2229875 | Zbl 1113.14011

[39] Sabbah, Claude Fourier-Laplace transform of a variation of polarized complex Hodge structure, J. Reine Angew. Math., Tome 621 (2008), pp. 123-158 | Article | MR 2431252 | Zbl 1155.32012

[40] Sabbah, Claude Wild twistor 𝒟-modules, Algebraic analysis and around, Math. Soc. Japan, Tokyo (Adv. Stud. Pure Math.) Tome 54 (2009), pp. 293-353 | MR 2499560 | Zbl 1219.32013

[41] Sabbah, Claude Fourier-Laplace transform of a variation of polarized complex Hodge structure, II, New developments in algebraic geometry, integrable systems and mirror symmetry (RIMS, Kyoto, 2008), Math. Soc. Japan, Tokyo (Adv. Stud. Pure Math.) Tome 59 (2010), pp. 289-347 | MR 2683213 | Zbl pre05872337

[42] Sabbah, Claude Introduction to Stokes structures (2010) (Lecture Notes (Lisboa, January 2009) 200 pages, arXiv: 0912.2762) | MR 2978128 | Zbl pre06062379

[43] Saito, Morihiko Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci., Tome 24 (1988) no. 6, p. 849-995 (1989) | Article | MR 1000123 | Zbl 0691.14007

[44] Saito, Morihiko Induced 𝒟-modules and differential complexes, Bull. Soc. Math. France, Tome 117 (1989) no. 3, pp. 361-387 | Numdam | MR 1020112 | Zbl 0705.32005

[45] Saito, Morihiko On the structure of Brieskorn lattice, Ann. Inst. Fourier (Grenoble), Tome 39 (1989) no. 1, pp. 27-72 | Numdam | MR 1011977 | Zbl 0644.32005

[46] Saito, Morihiko Mixed Hodge modules, Publ. Res. Inst. Math. Sci., Tome 26 (1990) no. 2, pp. 221-333 | Article | MR 1047415 | Zbl 0727.14004

[47] Shklyarov, D. Non-commutative Hodge structures: towards matching categorical and geometric examples (2011) (arXiv: 1107.3156)

[48] Simpson, Carlos T. Harmonic bundles on noncompact curves, J. Amer. Math. Soc., Tome 3 (1990) no. 3, pp. 713-770 | Article | MR 1040197 | Zbl 0713.58012

[49] Simpson, Carlos T. Nonabelian Hodge theory, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo (1991), pp. 747-756 | MR 1159261 | Zbl 0765.14005

[50] Simpson, Carlos T. Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. (1992) no. 75, pp. 5-95 | Numdam | MR 1179076 | Zbl 0814.32003

[51] Simpson, Carlos T. The Hodge filtration on nonabelian cohomology, Algebraic geometry—Santa Cruz 1995, Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 62 (1997), pp. 217-281 | MR 1492538 | Zbl 0914.14003

[52] Simpson, Carlos T. Mixed twistor structures (1997) (Prépublication Université de Toulouse & arXiv: math.AG/9705006)

[53] Simpson, Carlos T. Algebraic aspects of higher nonabelian Hodge theory, Motives, polylogarithms and Hodge theory, Part II (Irvine, CA, 1998), Int. Press, Somerville, MA (Int. Press Lect. Ser.) Tome 3 (2002), pp. 417-604 | MR 1978713 | Zbl 1051.14008

[54] Steenbrink, J. H. M. The spectrum of hypersurface singularities, Astérisque (1989) no. 179-180, pp. 11, 163-184 (Actes du Colloque de Théorie de Hodge (Luminy, 1987)) | MR 1042806 | Zbl 0725.14031

[55] Varchenko, A.N. Asymptotic Hodge structure on the cohomology of the Milnor fiber, Math. USSR Izv., Tome 18 (1982), pp. 469-512 | Zbl 0489.14003