Non-commutative Hodge structures

Sabbah, Claude

Annales de l'Institut Fourier, Tome 61 (2011) no. 7, pp. 2681-2717.

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*.

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.

Classification : 14D07, 34M40
Keywords: Non-commutative Hodge structure, Fourier-Laplace transformation, Brieskorn lattice
Mot clés : Structure de Hodge non commutative, transformation de Fourier-Laplace, réseau de Brieskorn

[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., Volume 12 (1981) no. 5, pp. 691-721 | DOI | MR | Zbl

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

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

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

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

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

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

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

[9] Hertling, Claus Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts in Mathematics, 151, Cambridge University Press, Cambridge, 2002 | DOI | MR | Zbl

[10] Hertling, Claus $t{t}^{*}$ geometry, Frobenius manifolds, their connections, and the construction for singularities, J. Reine Angew. Math., Volume 555 (2003), pp. 77-161 | DOI | MR | Zbl

[11] Hertling, Claus $t{t}^{*}$ geometry and mixed Hodge structures, Singularity theory and its applications (Adv. Stud. Pure Math.), Volume 43, Math. Soc. Japan, Tokyo, 2006, pp. 73-84 | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

[34] Peters, Chris A. M.; Steenbrink, Joseph H. M. Mixed Hodge structures, 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], 52, Springer-Verlag, Berlin, 2008 | MR | Zbl

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

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

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

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

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

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

[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) (Adv. Stud. Pure Math.), Volume 59, Math. Soc. Japan, Tokyo, 2010, pp. 289-347 | MR

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

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

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

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

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

[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., Volume 3 (1990) no. 3, pp. 713-770 | DOI | MR | Zbl

[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 | Zbl

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

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

[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 Lect. Ser.), Volume 3, Int. Press, Somerville, MA, 2002, pp. 417-604 | MR | Zbl

[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) | Numdam | MR | Zbl

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