Abstract
In this paper, we construct extensions of mixed Hodge structure coming from the mixed Hodge structure on the graded quotients of the group ring of the fundamental group of a smooth projective pointed curve which correspond to the regulators of certain motivic cohomology cycles on the Jacobian of the curve essentially constructed by Bloch and Beilinson. This leads to a new iterated integral expression for the regulator. This is a generalisation of a theorem of Colombo (J. Algebr. Geom. 11(4) (2002) 761–790) where she constructed the extension corresponding to Collino’s cycles in the Jacobian of a hyperelliptic curve.
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Acknowledgements
This work constitutes part of the Ph.D. thesis of the first author. The authors would like to thank Najmuddin Fakhruddin, Noriyuki Otsubo, Satoshi Kondo, Elisabetta Colombo, Jishnu Biswas, Manish Kumar, Ronnie Sebastian, Ranier Kaenders, Harish Seshadri, Arvind Nair, Shreedhar Inamdar and Suresh Nayak for their comments and suggestions. They would also like to thank the referee of an earlier version of this manuscript for pointing out numerous errors and specifically for pointing out the error in Colombo’s paper which we had to rectify and the referee of this version for their comments and suggestions. Finally, the authors take great pleasure in thanking the Indian Statistical Institute, Bangalore for their support while this work was being done. The first author would also like to thank TIFR for their hospitality while these revisions were being made.
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Sarkar, S., Sreekantan, R. The fundamental group and extensions of motives of Jacobians of curves. Proc Math Sci 130, 18 (2020). https://doi.org/10.1007/s12044-019-0539-z
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DOI: https://doi.org/10.1007/s12044-019-0539-z
Keywords
- Algebraic cycles
- mixed Hodge structures
- extensions
- regulators
- curves
- Jacobians
- higher Chow cycles
- motivic cycles