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Mathematics  2015 

Algebraic Cycles and Extensions of Mixed Hodge Structures Arising from the Fundamental Group of a Punctured Curve

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Abstract:

Let $X$ be a smooth projective curve over $\mathbb{C}$, $\infty,p\in X(\mathbb{C})$, and $L_n$ be the mixed Hodge structure of functions on $\pi_1(X-\{\infty\},p)$ that can be expressed by iterated integrals of length at most $n$. In this paper we express the mixed Hodge extension $e^\infty_{n,p}$ arisen from the weight filtration on $\displaystyle{\frac{L_n}{L_{n-2}}}$ in terms of certain null-homologous algebraic cycles on $X^{2n-1}$. In the final section, we assume $X, p,\infty$ are over a subfield $k\subset\mathbb{C}$ and show that, one can associate to the extension $e^\infty_{n,p}$ a family of $k$-valued points on the Jacobian of $X$, parametrized by elements of $\text{CH}_{n-1}(X^{2n-1})$ defined over $k$. When $n=2$, the results are due to Darmon, Rotger and Sols. Our contribution is in generalizing the picture to $n\geq3$.

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