Illinois Journal of Mathematics

Kuranishi spaces of meromorphic connections

Francois-Xavier Machu

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We construct the Kuranishi spaces, or in other words, the versal deformations, for the following classes of connections with fixed divisor of poles $D$: all such connections, as well as for its subclasses of integrable, integrable logarithmic and integrable logarithmic connections with a parabolic structure over $D$. The tangent and obstruction spaces of deformation theory are defined as the hypercohomology of an appropriate complex of sheaves, and the Kuranishi space is a fiber of the formal obstruction map.

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Illinois J. Math., Volume 55, Number 2 (2011), 509-541.

First available in Project Euclid: 1 February 2013

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Primary: 14B12: Local deformation theory, Artin approximation, etc. [See also 13B40, 13D10] 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] 14F40: de Rham cohomology [See also 14C30, 32C35, 32L10] 14H60: Vector bundles on curves and their moduli [See also 14D20, 14F05] 32G08: Deformations of fiber bundles


Machu, Francois-Xavier. Kuranishi spaces of meromorphic connections. Illinois J. Math. 55 (2011), no. 2, 509--541. doi:10.1215/ijm/1359762400.

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  • V. I. Arnol'd, S. M. Gusejn-Zade and A. N. Varchenko, Singularities of differentiable maps. Volume I: The classification of critical points, caustics and wave fronts, Monographs in Mathematics, vol. 82, Birkhäuser, Boston, 1985.
  • I. V. Artamkin, On deformation of sheaves, Izv. Akad. Nauk SSSR 52 (1988), 660–665.
  • I. V. Artamkin, Deforming torsion-free sheaves on algebraic surfaces, Izv. Akad. Nauk SSSR 54 (1990), 435–468.
  • M. Artin, Lectures on deformations of singularities, Lectures on Mathematics and Physics, vol. 54, Tata Institute of Fundamental Research, Bombay, 1976, 127 pp. Notes by C. S. Seshadri and A. Tannenbaum.
  • J. Bingener, Lokale Modulräume in der analytischen Geometrie. Band 1 und 2, Aspekte der Mathematik, D2, D3, Vieweg & Sohn, Braunschweig, 1987.
  • I. Biswas and S. Ramanan, An infinitesimal study of the moduli of Hitchin pairs, J. Lond. Math. Soc. (2) 49 (1994), 219–231.
  • H. Esnault and E. Viehweg, Logarithmic de Rham complexes and vanishing theorems, Invent. Math. 86 (1986), 161–194.
  • B. Fantechi and M. Manetti, Obstruction calculus for functors of Artin rings. I, J. Algebra 202 (1998), 541–576.
  • H. Flenner, Deformationen holomorpher Abbildungen, Habilitationsschrift, Osnabrück, 1978.
  • A. Grothendieck, Descent techniques and existence theorems in algebraic geometry. Picard's schemes: Existence theorems, Séminaire Bourbaki, vol. 7, Exp. no. 232, Soc. Math. France, Paris, 1995, pp. 143–161.
  • D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997.
  • L. Illusie, Complexe cotangent et déformations, I, Lecture Notes in Mathematics, vol. 239, Springer-Verlag, Berlin, 1972. (In French.)
  • L. Illusie, Complexe cotangent et déformations, II, Lecture Notes in Mathematics, vol. 283, Springer-Verlag, Berlin, 1973. (In French.)
  • M-A. Inaba, Moduli of parabolic connections on a curve and Riemann–Hilbert correspondence, available at \arxivurlmath.AG/0602004.
  • K. Kodaira and D. C. Spencer, On deformations of complex analytic structures, I, II, Ann. of Math. (2) 67 (1958), 328–466.
  • K. Kodaira and D. C. Spencer, On deformations of complex analytic structures, III, Stability theorems for complex structures, Ann. of Math. (2), 71 (1963), 43–76.
  • M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), 157–216.
  • M. Kuranishi, On deformations of compact complex structures, Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, pp. 357–359.
  • M. Kuranishi, On the locally complete families of complex analytic structures, Ann. of Math. (2) 75 (1962), 536–577.
  • M. Manetti, Deformation theory via differential graded algebra, Seminari di Geometria Algebrica 1998–1999, Scuola Normale di Pisa, 1999.
  • S. Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3-surface, Invent. Math. 77 (1984), 101–116.
  • N. Nitsure, Moduli space of semistable logarithmic connections, J. Amer. Math. Soc. 6 (1993), 597–609.
  • V. P. Palamodov, Deformations of complex spaces, Uspekhi Mat. Nauk 31 (1976), 129–194.
  • V. P. Palamodov, Deformations of complex spaces, Current problems in mathematics, Fundamental directions, vol. 10, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1986, pp. 123–221.
  • Z. Ran, Deformations of maps, Algebraic curves and projective geometry (Trento, 1988), Lecture Notes in Math., vol. 1389, Springer, Berlin, 1989, pp. 246–253.
  • Z. Ran, Hodge theory and deformations of maps, Compositio Math. 97 (1995), 309–328.
  • D. S. Rim, Formal Deformation Theory, SGA 7, Exposé VI, Lect. Notes Math., vol. 288, Springer-Verlag, Berlin, 1972.
  • D. S. Rim, Equivariant $G$-structure on versal deformations, Trans. Amer. Soc. 257 (1980), 217–226.
  • M. Schlessinger, Functor of Artin rings, Trans. Amer. Math. Soc. 30 (1968), 208–222.
  • M. Schlessinger, On rigid singularities, Rice Univ. Studies 59 (1973), 147–162.