Algebraic cycles and the mixed Hodge structure on the fundamental group of a punctured curve

Abstract

Let X be a smooth projective curve of genus $$\ge 1$$ over $${{\mathbb {C}}}$$ , and $$e,\infty \in X({{\mathbb {C}}})$$ be distinct points. Let $$L_n$$ be the mixed Hodge structure of functions on $$\pi _1(X-\{\infty \},e)$$ given by iterated integrals of length $$\le n$$ (as defined by Hain). Building on a work of Darmon et al. (Publications Mathematiques de Besancon 2:19–46, 2012), we express the mixed Hodge extension $${\mathbb {E}}^\infty _{n,e}$$ given by the weight filtration on $$\frac{L_n}{L_{n-2}}$$ as the Abel–Jacobi image of a null-homologous algebraic cycle on $$X^{2n-1}$$ . This algebraic cycle is constructed using the different embeddings of $$X^{n-1}$$ into $$X^n$$ . As a corollary, we show that the extension $${\mathbb {E}}^\infty _{n,e}$$ determines the point $$\infty \in X-\{e\}$$ . When $$n=2$$ , our main result is a strengthening of a theorem of Darmon et al. (Publications Mathematiques de Besancon 2:19–46, 2012). In the final section we assume that $$X,e,\infty $$ are defined over a subfield K of $${{\mathbb {C}}}$$ . Generalizing a construction in (Darmon et al., Publications Mathematiques de Besancon 2:19–46, 2012), we use the extension $${\mathbb {E}}^\infty _{n,e}$$ to define a family of K-rational points on the Jacobian of X parametrized by $$(n-1)$$ -dimensional algebraic cycles on $$X^{2n-2}$$ defined over K.