The mixed Hodge structure on the fundamental group of hyperelliptic curves and higher cycles

Abstract

In this paper we give a geometrical interpretation of an extension of mixed Hodge structures (MHS) obtained from the canonical MHS on the group ring of the fundamental group of a hyperelliptic curve modulo the fourth power of its augmentation ideal. We show that the class of this extension coincides with the regulator image of a canonical higher cycle in a hyperelliptic Jacobian. This higher cycle was introduced and studied by Collino.

The fundamental group and extensions of motives of Jacobians of curves

“…The cycle is similar, in fact, generalises, the cycle constructed by Collino [Col97]. This section generalises the work of Colombo [Col02] on constructing the extension corresponding to the Collino cycle and hence many of the arguments are adapted from her paper.…”

“…Darmon-Rotger-Sols [DRS12] have used the modified diagonal cycle to construct points on Jacobians of the curves and used the iterated integral approach to find a formula for the Abel-Jacobi image of these points. Collino [Col97] and Colombo [Col02] it has been known that these null homologous cycles degenerate to higher Chow cycles on related varieties. Recently Iyer and Müller-Stach [IM14] have shown that the modified diagonal cycle degenerates to the kind of cycles we consider in some special cases.…”

“…and one can project to get two classes e 4 Q,P and e 4 R,P in Ext(⊗ 3 H 1 (C), H 1 (C)). Colombo [Col02] shows that the class e 4 QR,P = e 4 Q,P ⊖ B e 4 R,P ∈ Ext(⊗ 3 H 1 (C), H 1 (C)) corresponds to the extension determined by the cycle Z QR -after pulling back and pushing forward with some standard maps.…”

Iterated Integrals and Algebraic Cycles: Examples and Prospects

“…If C is hyperelliptic and x and y are two distinct Weierstrass points, the mixed Hodge structure on J(C − {y}, x)/J 3 is of order 2. In this case Colombo [19] constructs an extension of Z by the primitive part P H 2 (Jac C; Z) of H 2 (Jac C) from the MHS on J(C − {y}, x)/J 4 and shows that it is the class of the Collino cycle [18], an element of the Bloch higher Chow group CH g (Jac C, 1). This example shows that the MHS on π 1 (C − {y}, x) of a hyperelliptic curve contains information about the extensions associated to elements of higher K-groups, (K 1 in this case), not just K 0 .…”

“…Examples relating iterated integrals and motives go back to Bloch's work on the dilogarithm and regulators [9] in the mid 1970s, which was developed further by Beilinson [7] and Deligne (unpublished). Further evidence to support a connection between de Rham homotopy theory and iterated integrals includes [48,4,30,57,50,22,37,59,58,25,38,51,55,19,60,20]. Chen would have been delighted by these developments, as he believed iterated integrals and loopspaces contain non-trivial geometric information and would one day become a useful mathematical tool outside topology.The paper is largely expository, beginning with an introduction to iterated integrals and Chen's de Rham theorems for loop spaces and fundamental groups.…”