Semi-algebraic chains on projective varieties and the Abel-Jacobi map for higher Chow cycles

Kimura, Kenichiro

doi:10.1090/tran/8422

Semi-algebraic chains on projective varieties and the Abel-Jacobi map for higher Chow cycles
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Trans. Amer. Math. Soc. 374 (2021), 7589-7619 Request permission

Abstract:

We will show that the singular cohomology groups of a smooth quasi-projective complex variety relative to a normal crossing divisor can be described in terms of $\delta$-admissible chains. Roughly speaking, a $\delta$-admissible chain is a simplicial semi-algebraic chain meeting the “faces” properly. As an application, we show that the Abel-Jacobi map for higher Chow cycles can be described via $\delta$-admissible chains. As an example, we will describe the Hodge realization of the polylog cycles constructed by Bloch-Kriz in terms of the Abel-Jacobi map.

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