## Journal of the Mathematical Society of Japan

### Moduli of regular singular parabolic connections with given spectral type on smooth projective curves

#### Abstract

We define a moduli space of stable regular singular parabolic connections with given spectral type on smooth projective curves and show the smoothness of the moduli space and give a relative symplectic structure on the moduli space. Moreover, we define the isomonodromic deformation on this moduli space and prove the geometric Painlevé property of the isomonodromic deformation.

#### Note

This work was partly supported by JSPS KAKENHI: Grant Numbers JP17H06127, JP15K13427, JP24224001, JP22740014, JP26400043.

#### Article information

Source
J. Math. Soc. Japan, Volume 70, Number 3 (2018), 879-894.

Dates
First available in Project Euclid: 31 May 2018

https://projecteuclid.org/euclid.jmsj/1527795357

Digital Object Identifier
doi:10.2969/jmsj/76597659

Mathematical Reviews number (MathSciNet)
MR3832081

Zentralblatt MATH identifier
06966965

#### Citation

INABA, Michi-aki; SAITO, Masa-Hiko. Moduli of regular singular parabolic connections with given spectral type on smooth projective curves. J. Math. Soc. Japan 70 (2018), no. 3, 879--894. doi:10.2969/jmsj/76597659. https://projecteuclid.org/euclid.jmsj/1527795357

#### References

• W. Crawley-Boevey, Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity, Publ. Math. Inst. Hautes Études Sci., 100 (2004), 171–207.
• K. Fuji and T. Suzuki, Drinfeld–Sokolov hierarchies of type A and fourth order Painlevé system, Funkcial. Ekvac., 53 (2010), 143–167.
• K. Fuji and T. Suzuki, Higher order Painlevé system of type $D^{(1)}_{2n+2}$ arising from integrable hierarchy, Int. Math. Res. Not. IMRN, 2008 (2008), Art.ID rnm129.
• M. Inaba, Moduli of parabolic connections on a curve and Riemann–Hilbert correspondence, J. Algebraic Geom., 22 (2013), 407–480.
• M. Inaba, K. Iwasaki and M.-H. Saito, Moduli of stable parabolic connections, Riemann–Hilbert correspondence and geometry of painlevé equation of type VI, Part I, Publ. Res. Inst. Math. Sci., 42 (2006), 987–1089.
• M. Inaba, K. Iwasaki and M.-H. Saito, Dynamics of the sixth Pailevé Equations, in Théories asymptotiques et équations de Painlevé, Sémin. Congr., 14, Soc. Math. France, Paris, 2006, 103–167.
• H. Kawakami, A. Nakamura and H. Sakai, Degeneration scheme of 4-dimensional Painlevé equations, arXiv:1209.3836.
• H. Kawakami, A. Nakamura and H. Sakai, Toward a classification of four-dimensional Painlevé-type equations, Algebraic and geometric aspects of integrable systems and random matrices, Contemp. Math., 593, Amer. Math. Soc., Providence, RI, 2013, 143–161.
• T. Oshima, Classification of Fuchsian systems and their connection problem, Exact WKB analysis and microlocal analysis, RIMS Kôkyûroku Bessatsu, B37, Res. Inst. Math. Sci. (RIMS), Kyoto, 2013, arXiv:0811.2916v2, 163–192.
• M.-H. Saito and S. Szabo, Apparent singularities and canonical coordinates for moduli of parabolic connections and parabolic Higgs bundles, in preparation.
• H. Sakai, Isomonodromic deformation and $4$-dimensional Painlevé type equations, preprint, University of Tokyo, Mathematical Sciences, 2010.
• Y. Sasano, Coupled Painlevé VI systems in dimension four with affine Weyl group symmetry of type $D^{(1)}_6$, II, RIMS Kôkyûroku Bessatsu B5, 2008, 137–152.
• D. Yamakawa, Geometry of multiplicative preprojective algebra, Int. Math. Res. Pap. IMRP 2008, Art. ID rpn008, 77pp. (arXiv:0710.10.2649v4 [math.SG] 15 Nov 2007)