Speaker
Description
For $\mathsf{P}$ smooth projective variety of dim. $n$ over $\mathbb{C}$ complex numbers; $Y=Y_1-Y_2 $codim r algebraic cycle, $Z=Z_1-Z_2$ dim $r-1$ algebraic cycle, $Y$, $Z$ disjoint support and homologous to 0. Biextension $\mathsf{B} := H^{2r-1}(\mathsf{P}\smallsetminus Y,Z;\mathbb{Q}(r))$ mixed $\mathbb{Q}_{HS}$ with weights 0,-1,-2 and weight graded $W_{-2}\mathsf{B}=\mathbb{Q}(1)$, $gr^W_{-1}\mathsf{B}=H^{2r-1}(\mathsf{P},\mathbb{Q}(r))$, and $gr^W_0 = \mathbb{Q}(0)$. Degenerate case $gr^W_{-1}\mathsf{B}=(0)$ yields a Kummer extension $0 \to \mathbb{Z}(1) \to \mathsf{B} \to \mathbb{Z}(0)\to 0$. Such a Kummer extension carries a generalized cross-ratio $\lambda(\mathsf{B}) \in \mathbb{C}^*$. Examples and conjectures about generalized cross-ratios will be discussed.
