The fundamental group and extensions of motives of Jacobians of curves
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The fundamental group and extensions of motives of Jacobians of curves SUBHAM SARKAR1 and RAMESH SREEKANTAN2,∗ 1 School of Mathematics, Tata Institute of Fundamental Research, Dr. Homi Bhabha Road, Colaba, Mumbai 400 005, India 2 Statistics and Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, Jnanabharathi, Bengaluru 560 059, India *Corresponding author. E-mail: [email protected]; [email protected]
MS received 6 September 2018; revised 21 May 2019; accepted 29 May 2019 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. Keywords. Algebraic cycles; mixed Hodge structures; extensions; regulators; curves; Jacobians; higher Chow cycles; motivic cycles. Mathematics Subject Classification.
19F27, 11G55, 14C30, 14C35.
1. Introduction A formula, usually called Beilinson’s formula – though independently due to Deligne as well – describes the motivic cohomology group of a smooth projective variety X over a number field as the group of extensions in a conjectured category of mixed motives, MMQ . If i and n are two integers, then [19] Ext MMQ (Q(−n), h (X )) = 1
i
n (X ) ⊗ Q C Hhom i+1
HM (X, Q(n))
if i + 1 = 2n, if i + 1 = 2n.
Hence, if one had a way of constructing extensions in the category of mixed motives by some other method, it would provide a way of constructing motivic cycles. One way of doing so is by considering the group ring of the fundamental group of the algebraic variety Z[π1 (X, P)]. If J P is the augmentation ideal – the kernel of the map from Z[π1 (X, P)] → Z – then the graded pieces J Pa /J Pb with a < b are expected to have © Indian Academy of Sciences 0123456789().: V,-vol
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Proc. Indian Acad. Sci. (Math. Sci.)
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a motivic structure. These give rise to natural extensions of motives – so one could hope that these extensions could be used to construct natural motivic cycles. Understanding the motivic structure on the fundamental group is complicated. However, the Hodge structure on the fundamental group is well understood [9]. The regulator of a motivic cohomology cycle can be thought of as the realisation of the corresponding extension of motives as an extension in the category of mixed Hodge structures. So while we may not be able to construct motivic cycles as extensions of motives coming from the fundamental group – we can hope to construct their regulators as extensions of mixed Hodge structures (MHS) coming from the fundamental group. The aim of this
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