Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 082, 73 pages      arXiv:2009.14314
Contribution to the Special Issue on Primitive Forms and Related Topics in honor of Kyoji Saito for his 77th birthday

Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1

Claus Hertling
Lehrstuhl für algebraische Geometrie, Universität Mannheim, B6, 26, 68159 Mannheim, Germany

Received September 30, 2020, in final form August 20, 2021; Published online September 07, 2021

Holomorphic vector bundles on $\mathbb C\times M$, $M$ a complex manifold, with meromorphic connections with poles of Poincaré rank 1 along $\{0\}\times M$ arise naturally in algebraic geometry. They are called $(TE)$-structures here. This paper takes an abstract point of view. It gives a complete classification of all $(TE)$-structures of rank 2 over germs $\big(M,t^0\big)$ of manifolds. In the case of $M$ a point, they separate into four types. Those of three types have universal unfoldings, those of the fourth type (the logarithmic type) not. The classification of unfoldings of $(TE)$-structures of the fourth type is rich and interesting. The paper finds and lists also all $(TE)$-structures which are basic in the following sense: Together they induce all rank $2$ $(TE)$-structures, and each of them is not induced by any other $(TE)$-structure in the list. Their base spaces $M$ turn out to be 2-dimensional $F$-manifolds with Euler fields. The paper gives also for each such $F$-manifold a classification of all rank 2 $(TE)$-structures over it. Also this classification is surprisingly rich. The backbone of the paper are normal forms. Though also the monodromy and the geometry of the induced Higgs fields and of the bases spaces are important and are considered.

Key words: meromorphic connections; isomonodromic deformations; $(TE)$-structures.

pdf (1027 kb)   tex (77 kb)  


  1. Anosov D.V., Bolibruch A.A., The Riemann-Hilbert problem, Aspects of Mathematics, Vol. 22, Friedr. Vieweg & Sohn, Braunschweig, 1994.
  2. Babbitt D.G., Varadarajan V.S., Formal reduction theory of meromorphic differential equations: a group theoretic view, Pacific J. Math. 109 (1983), 1-80.
  3. Cotti G., Dubrovin B., Guzzetti D., Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, Duke Math. J. 168 (2019), 967-1108, arXiv:1706.04808.
  4. David L., Hertling C., Regular $F$-manifolds: initial conditions and Frobenius metrics, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), 1121-1152, arXiv:1411.4553.
  5. David L., Hertling C., $(T)$-structures over two-dimensional $F$-manifolds: formal classification, Ann. Mat. Pura Appl. (4) 199 (2020), 1221-1242, arXiv:1811.03406.
  6. David L., Hertling C., $(TE)$-structures over the irreducible 2-dimensional globally nilpotent $F$-manifold germ, Rev. Roumaine Math. Pures Appl. 65 (2020), 235-284, arXiv:2001.01063.
  7. David L., Hertling C., Meromorphic connections over F-manifolds, in Integrability, Quantization, and Geometry. I. Integrable systems, Proceedings of Symposia in Pure Mathematics, Vol. 103, Editors S. Novikov, I. Krichever, O. Ogievetsky, S. Shlosman, Amer. Math. Soc., Providence, RI, 2021, 171-216, arXiv:1912.03331.
  8. Fabry E., Sur les intégrales des équations différentielles lineaires à coefficients rationels, Ph.D. Thesis, Paris, 1885.
  9. Hertling C., Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts in Mathematics, Vol. 151, Cambridge University Press, Cambridge, 2002.
  10. Hertling C., $tt^*$ geometry, Frobenius manifolds, their connections, and the construction for singularities, J. Reine Angew. Math. 555 (2003), 77-161, arXiv:math.AG/0203054.
  11. Hertling C., Hoevenaars L., Posthuma H., Frobenius manifolds, projective special geometry and Hitchin systems, J. Reine Angew. Math. 649 (2010), 117-165, arXiv:0905.3304.
  12. Hertling C., Manin Yu., Weak Frobenius manifolds, Int. Math. Res. Not 1999 (1999), 277-286, arXiv:math.QA/9810132.
  13. Hertling C., Manin Yu., Unfoldings of meromorphic connections and a construction of Frobenius manifolds, in Frobenius Manifolds, Aspects Math., Vol. 36, Friedr. Vieweg, Wiesbaden, 2004, 113-144, arXiv:math.AG/0207089.
  14. Hertling C., Sabbah C., Examples of non-commutative Hodge structures, J. Inst. Math. Jussieu 10 (2011), 635-674, arXiv:0912.2754.
  15. Hertling C., Sevenheck C., Limits of families of Brieskorn lattices and compactified classifying spaces, Adv. Math. 223 (2010), 1155-1224, arXiv:0805.4777.
  16. Malgrange B., La classification des connexions irrégulières à une variable, in Mathematics and Physics (Paris, 1979/1982), Progr. Math., Vol. 37, Birkhäuser Boston, Boston, MA, 1983, 381-399.
  17. Malgrange B., Sur les déformations isomonodromiques. I. II. Singularités irrégulières, in Mathematics and physics (Paris, 1979/1982), Progr. Math., Vol. 37, Birkhäuser Boston, Boston, MA, 1983, 401-438.
  18. Malgrange B., Deformations of differential systems. II, J. Ramanujan Math. Soc. 1 (1986), 3-15.
  19. Sabbah C., Introduction to algebraic theory of linear systems of differential equations, in Éléments de la théorie des systèmes différentiels. $\mathcal D$-modules cohérents et holonomes (Nice, 1990), Travaux en Cours, Vol. 45, Hermann, Paris, 1993, 1-80.
  20. Sabbah C., Polarizable twistor $\mathcal D$-modules, Astérisque 300 (2005), vi+208 pages, arXiv:math.AG/0503038.
  21. Sabbah C., Isomonodromic deformations and Frobenius manifolds. An introduction, Universitext, Springer, London, 2008.
  22. Sabbah C., Integrable deformations and degenerations of some irregular singularities, Publ. Res. Inst. Math. Sci., to appear, arXiv:1711.08514.
  23. Saito M., Deformations of abstract Brieskorn lattices, arXiv:1707.07480.
  24. Varadarajan V.S., Linear meromorphic differential equations: a modern point of view, Bull. Amer. Math. Soc. (N.S.) 33 (1996), 1-42.

Previous article  Next article  Contents of Volume 17 (2021)