Hostname: page-component-669899f699-swprf Total loading time: 0 Render date: 2025-04-27T03:26:55.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Examples of non-commutative Hodge structures

Part of: Families, fibrations

Published online by Cambridge University Press:  10 March 2011

Claus Hertling  and
Claude Sabbah
Claus Hertling
Affiliation:
Lehrstuhl für Mathematik VI, Universität Mannheim, Seminargebäude A 5, 6, 68131 Mannheim, Germany ([email protected])
Claude Sabbah
Affiliation:
UMR 7640 du CNRS, Centre de Mathématiques Laurent Schwartz, École polytechnique, 91128 Palaiseau cedex, France ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that, under a condition called minimality, if the Stokes matrix of a connection with a pole of order two and no ramification gives rise, when added to its adjoint, to a positive semidefinite Hermitian form, then the associated integrable twistor structure (or TERP structure, or noncommutative Hodge structure) is pure and polarized.

Keywords

Hermitian pairingLaplace transformationmeromorphic connectionStokes matrixnon-commutative Hodge structure

MSC classification

Secondary: 14D07: Variation of Hodge structures
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 10 , Special Issue 3: A Special Issue in Honour of Louis Boutet de Monvel and Pierre Schapira , July 2011 , pp. 635 - 674
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1.Babbitt, D. G. and Varadarajan, V. S., Local moduli for meromorphic differential equations, Astérisque, Volume 169–170 (Société Mathématique de France, Paris, 1989).Google Scholar
2.Cecotti, S. and Vafa, C., Topological-antitopological fusion, Nucl. Phys. B 367 (1991), 359–461.CrossRefGoogle Scholar
3.Cecotti, S. and Vafa, C., On classification of N = 2 supersymmetric theories, Commun. Math. Phys. 158 (1993), 569–644.CrossRefGoogle Scholar
4.Cecotti, S., Fendley, P., Intriligator, K. and Vafa, C., A new supersymmetric index, Nucl. Phys. B386 (1992), 405–452.CrossRefGoogle Scholar
5.Deligne, P., Lettre à B. Malgrange du 19/4/1978, in Singularités irrégulières: correspondence et documents, Documents Mathématiques, Volume 5, pp. 25–26 (Société Mathématique de France, Paris, 2007).Google Scholar
6.Douai, A. and Sabbah, C., Gauss-Manin systems, Brieskorn lattices and Frobenius structures (I), Annales Inst. Fourier 53(4) (2003), 1055–1116.CrossRefGoogle Scholar
7.Godement, R., Topologie algébrique et théorie des faisceaux (Hermann, Paris, 1964).Google Scholar
8.Hertling, C., Classifying spaces for polarized mixed Hodge structures and for Brieskorn lattices, Compositio Math. 116 (1) (1999), 1–37.CrossRefGoogle Scholar
9.Hertling, C., tt* geometry, Frobenius manifolds, their connections, and the construction for singularities, J. Reine Angew. Math. 555 (2003), 77–161.Google Scholar
10.Hertling, C., tt* geometry and mixed Hodge structures, in Singularity theory and its applications, Advanced Studies in Pure Mathematics, Volume 43, pp. 73–84 (Mathematical Society of Japan, Tokyo, 2006).Google Scholar
11.Hertling, C. and Sevenheck, Ch., Nilpotent orbits of a generalization of Hodge structures, J. Reine Angew. Math. 609 (2007), 23–80.Google Scholar
12.Hertling, C. and Sevenheck, Ch., Curvature of classifying spaces for Brieskorn lattices, J. Geom. Phys. 58 (11) (2008), 1591–1606.CrossRefGoogle Scholar
13.Hertling, C. and Sevenheck, Ch., Twistor structures, tt*-geometry and singularity theory, in From Hodge theory to integrability and TQFT: tt*-geometry (ed. Donagi, R. and Wendland, K.), Proceedings of Symposia in Pure Mathematics, Volume 78, pp. 49–73 (American Mathematical Society, Providence, RI, 2008).Google Scholar
14.Hertling, C. and Sevenheck, Ch., Limits of families of Brieskorn lattices and compactified classifying spaces, Adv. Math. 223 (2010), 1155–1224.CrossRefGoogle Scholar
15.Iritani, H., An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (3) (2009), 1016–1079.CrossRefGoogle Scholar
16.Iritani, H., tt*-geometry in quantum cohomology, preprint (arXiv: 0906.1307; 2009).Google Scholar
17.Kaledin, D., Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie, Pure Appl. Math. Q. 4(3) (2008), 785–875 (Special Issue in Honor of F. Bogomolov, Part 2).CrossRefGoogle Scholar
18.Kashiwara, M., Regular holonomic D-modules and distributions on complex manifolds, in Complex analytic singularities, Advanced Studies in Pure Mathematics, Volume 8, pp. 199–206 (North-Holland, Amsterdam, 1987).Google Scholar
19.Kashiwara, M. and Schapira, P., Sheaves on manifolds, Grundlehren der mathematischen Wissenschaften, Volume 292 (Springer, 1990).Google Scholar
20.Katzarkov, L., Kontsevich, M. and Pantev, T., Hodge theoretic aspects of mirror symmetry, in From Hodge theory to integrability and TQFT: tt*-geometry (ed. Donagi, R. and Wendland, K.), Proceedings of Symposia in Pure Mathematics, Volume 78, pp. 87–174 (American Mathematical Society, Providence, RI, 2008).Google Scholar
21.Malgrange, B., La classification des connexions irrégulières à une variable, in Séminaire E.N.S. Mathématique et Physique (ed. de Monvel, L. Boutet, Douady, A. and Verdier, J.-L.), Progress in Mathematics, Volume 37, pp. 381–399 (Birkhäuser, 1983).Google Scholar
22.Malgrange, B., Équations différentielles à coefficients polynomiaux, Progress in Mathematics, Volume 96 (Birkhäuser, 1991).Google Scholar
23.Malgrange, B., Connexions méromorphes, II, Le réseau canonique, Invent. Math. 124 (1996), 367–387.CrossRefGoogle Scholar
24.Malgrange, B., On irregular holonomic D-modules, in Éléments de la théorie des systèmes différentiels géométriques, Séminaires & Congrès, Volume 8, pp. 391–410 (Société Mathématique de France, Paris, 2004).Google Scholar
25.Mochizuki, T., Asymptotic behaviour of variation of pure polarized TERP structures, preprint (arXiv: 0811.1384; 2008).Google Scholar
26.Mochizuki, T., Wild harmonic bundles and wild pure twistor D-modules, preprint (arXiv: 0803.1344; 2008).Google Scholar
27.Sabbah, C., Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2, Astérisque, Volume 263 (Société Mathématique de France, Paris, 2000).Google Scholar
28.Sabbah, C., Polarizable twistor D-modules, Astérisque, Volume 300 (Société Mathématique de France, Paris, 2005).Google Scholar
29.Sabbah, C., Hypergeometric periods for a tame polynomial, Portugaliae Math. 63(2) (2006), 173–226.Google Scholar
30.Sabbah, C., Wild twistor D-modules, in Algebraic Analysis and Around: In Honor of Professor Masaki Kashiwara's 60th Birthday (Kyoto, June 2007), Advanced Studies in Pure Mathematics, Volume 54, pp. 293–353 (Mathematical Society of Japan, Tokyo, 2009).Google Scholar
31.Sabbah, C., Fourier-Laplace transform of a variation of polarized complex Hodge structure, J. Reine Angew. Math. 621 (2008), 123–158.Google Scholar
32.Sabbah, C., Fourier-Laplace transform of a variation of polarized complex Hodge structure, II, in New Developments in Algebraic Geometry, Integrable Systems and Mirror symmetry (Kyoto, January 2008), Advanced Studies in Pure Mathematics, Volume 59, pp. 289–347 (Mathematical Society of Japan, Tokyo, 2010).Google Scholar
33.Simpson, C., Mixed twistor structures, Prépublication, Université de Toulouse (arXiv: math.AG/9705006; 1997).Google Scholar