The Gauss-Manin connection for nonabelian cohomology spaces is the isomonodromy flow. We write down explicitly the vector fields of the isomonodromy flow and calculate its induced vector fields on the associated graded space of the nonabelian Hogde filtration. The result turns out to be intimately related to the quadratic part of the Hitchin map.
@article{bwmeta1.element.doi-10_2478_s11533-011-0110-3,
author = {Ting Chen},
title = {The associated map of the nonabelian Gauss-Manin connection},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {1407-1421},
zbl = {1282.14059},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0110-3}
}
Ting Chen. The associated map of the nonabelian Gauss-Manin connection. Open Mathematics, Tome 10 (2012) pp. 1407-1421. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0110-3/
[1] Ben-Zvi D., Frenkel E., Geometric realization of the Segal-Sugawara construction, In: Topology, Geometry and Quantum Field Theory, London Math. Soc. Lecture Note Ser., 308, Cambridge University Press, Cambridge, 2004, 46–97 http://dx.doi.org/10.1017/CBO9780511526398.006 | Zbl 1170.17303
[2] Griffiths P.A., Periods of integrals on algebraic manifolds II. (Local study of the period mapping), Amer. J. Math., 1968, 90(5), 805–865 http://dx.doi.org/10.2307/2373485 | Zbl 0183.25501
[3] Hitchin N.J., The self-duality equations on a Riemann surface, Proc. London Math. Soc., 1987, 55(1), 59–126 http://dx.doi.org/10.1112/plms/s3-55.1.59 | Zbl 0634.53045
[4] Inaba M., Iwasaki K., Saito M.-H., Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part I, Publ. Res. Inst. Math. Sci., 2006, 42(4), 987–1089 http://dx.doi.org/10.2977/prims/1166642194 | Zbl 1127.34055
[5] Markman E., Spectral curves and integrable systems, Compositio Math., 1994, 93(3), 255–290 | Zbl 0824.14013
[6] Mumford D., Projective invariants of projective structures and applications, In: Proc. Internat. Congr. Mathematicians, Stockholm, 1962, Inst. Mittag-Leffler, Djursholm, 1963, 526–530 | Zbl 0154.20702
[7] Simpson C., The Hodge filtration on nonabelian cohomology, In: Algebraic Geometry, Santa Cruz, 1995, Proc. Sympos. Pure Math., 62(2), American Mathematical Society, Providence, 1997, 217–281