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Cepu
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Let $X$ be a comapct Riemann surface of genus $g$ and let $J\: : \: X\to \mathbb{C}^{g}/\Lambda$ be the Abel-Jacobi map. This map is a smooth embedding. Let $p\in X$ such that $J(p)=\Lambda$ and consider $$ J\: : \: X\setminus p \to (\mathbb{C}-\Lambda)/\Lambda. $$ Then $J^*$ induces a surjection in complex de Rham cohomology. $J$ is injective and hence is defined at the level of the (ordered) configuration space $$ \operatorname{Conf}_{n}(J)\: : \: \operatorname{Conf}_{n}(X\setminus p) \to \operatorname{Conf}_{n}((\mathbb{C}-\Lambda)/\Lambda). $$$$ \operatorname{Conf}_{l}(J)\: : \: \operatorname{Conf}_{l}(X\setminus p) \to \operatorname{Conf}_{l}((\mathbb{C}-\Lambda)/\Lambda). $$ Let $V_{1}\subset H^{2}(\operatorname{Conf}_{n}(X\setminus p), \mathbb{C})$$V_{1}\subset H^{2}(\operatorname{Conf}_{l}(X\setminus p), \mathbb{C})$ , resp. $V_{2}\subset H^{2}(\operatorname{Conf}_{n}((\mathbb{C}-\Lambda)/\Lambda), \mathbb{C})$$V_{2}\subset H^{2}(\operatorname{Conf}_{l}((\mathbb{C}-\Lambda)/\Lambda), \mathbb{C})$ be the space generated by Massey products $(a_{1}, \dots, a_{n})$ between degree $1$ elements, for $n\geq 2$. Does $\operatorname{Conf}_{n}(J)^{*}$$\operatorname{Conf}_{l}(J)^{*}$ induces a surjection $$ \operatorname{Conf}_{n}(J)^{*}\: : \: \mathbb{C}\oplus H^{1}(\operatorname{Conf}_{n}((\mathbb{C}-\Lambda)/\Lambda), \mathbb{C})\oplus V_{2}\to \mathbb{C}\oplus H^{1}(\operatorname{Conf}_{n}(X\setminus p), \mathbb{C})\oplus V_{1}? $$$$ \operatorname{Conf}_{l}(J)^{*}\: : \: \mathbb{C}\oplus H^{1}(\operatorname{Conf}_{l}((\mathbb{C}-\Lambda)/\Lambda), \mathbb{C})\oplus V_{2}\to \mathbb{C}\oplus H^{1}(\operatorname{Conf}_{l}(X\setminus p), \mathbb{C})\oplus V_{1}? $$ Notice that $\mathbb{C}$ represents the $0$-th cohomology group.

Let $X$ be a comapct Riemann surface of genus $g$ and let $J\: : \: X\to \mathbb{C}^{g}/\Lambda$ be the Abel-Jacobi map. This map is a smooth embedding. Let $p\in X$ such that $J(p)=\Lambda$ and consider $$ J\: : \: X\setminus p \to (\mathbb{C}-\Lambda)/\Lambda. $$ Then $J^*$ induces a surjection in complex de Rham cohomology. $J$ is injective and hence is defined at the level of the (ordered) configuration space $$ \operatorname{Conf}_{n}(J)\: : \: \operatorname{Conf}_{n}(X\setminus p) \to \operatorname{Conf}_{n}((\mathbb{C}-\Lambda)/\Lambda). $$ Let $V_{1}\subset H^{2}(\operatorname{Conf}_{n}(X\setminus p), \mathbb{C})$ , resp. $V_{2}\subset H^{2}(\operatorname{Conf}_{n}((\mathbb{C}-\Lambda)/\Lambda), \mathbb{C})$ be the space generated by Massey products $(a_{1}, \dots, a_{n})$ between degree $1$ elements, for $n\geq 2$. Does $\operatorname{Conf}_{n}(J)^{*}$ induces a surjection $$ \operatorname{Conf}_{n}(J)^{*}\: : \: \mathbb{C}\oplus H^{1}(\operatorname{Conf}_{n}((\mathbb{C}-\Lambda)/\Lambda), \mathbb{C})\oplus V_{2}\to \mathbb{C}\oplus H^{1}(\operatorname{Conf}_{n}(X\setminus p), \mathbb{C})\oplus V_{1}? $$ Notice that $\mathbb{C}$ represents the $0$-th cohomology group.

Let $X$ be a comapct Riemann surface of genus $g$ and let $J\: : \: X\to \mathbb{C}^{g}/\Lambda$ be the Abel-Jacobi map. This map is a smooth embedding. Let $p\in X$ such that $J(p)=\Lambda$ and consider $$ J\: : \: X\setminus p \to (\mathbb{C}-\Lambda)/\Lambda. $$ Then $J^*$ induces a surjection in complex de Rham cohomology. $J$ is injective and hence is defined at the level of the (ordered) configuration space $$ \operatorname{Conf}_{l}(J)\: : \: \operatorname{Conf}_{l}(X\setminus p) \to \operatorname{Conf}_{l}((\mathbb{C}-\Lambda)/\Lambda). $$ Let $V_{1}\subset H^{2}(\operatorname{Conf}_{l}(X\setminus p), \mathbb{C})$ , resp. $V_{2}\subset H^{2}(\operatorname{Conf}_{l}((\mathbb{C}-\Lambda)/\Lambda), \mathbb{C})$ be the space generated by Massey products $(a_{1}, \dots, a_{n})$ between degree $1$ elements, for $n\geq 2$. Does $\operatorname{Conf}_{l}(J)^{*}$ induces a surjection $$ \operatorname{Conf}_{l}(J)^{*}\: : \: \mathbb{C}\oplus H^{1}(\operatorname{Conf}_{l}((\mathbb{C}-\Lambda)/\Lambda), \mathbb{C})\oplus V_{2}\to \mathbb{C}\oplus H^{1}(\operatorname{Conf}_{l}(X\setminus p), \mathbb{C})\oplus V_{1}? $$ Notice that $\mathbb{C}$ represents the $0$-th cohomology group.

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Cepu
Cepu
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Jacobian and configuration space and massey products

Let $X$ be a comapct Riemann surface of genus $g$ and let $J\: : \: X\to \mathbb{C}^{g}/\Lambda$ be the Abel-Jacobi map. This map is a smooth embedding. Let $p\in X$ such that $J(p)=\Lambda$ and consider $$ J\: : \: X\setminus p \to (\mathbb{C}-\Lambda)/\Lambda. $$ Then $J^*$ induces a surjection in complex de Rham cohomology. $J$ is injective and hence is defined at the level of the (ordered) configuration space $$ \operatorname{Conf}_{n}(J)\: : \: \operatorname{Conf}_{n}(X\setminus p) \to \operatorname{Conf}_{n}((\mathbb{C}-\Lambda)/\Lambda). $$ Let $V_{1}\subset H^{2}(\operatorname{Conf}_{n}(X\setminus p), \mathbb{C})$ , resp. $V_{2}\subset H^{2}(\operatorname{Conf}_{n}((\mathbb{C}-\Lambda)/\Lambda), \mathbb{C})$ be the space generated by Massey products $(a_{1}, \dots, a_{n})$ between degree $1$ elements, for $n\geq 2$. Does $\operatorname{Conf}_{n}(J)^{*}$ induces a surjection $$ \operatorname{Conf}_{n}(J)^{*}\: : \: \mathbb{C}\oplus H^{1}(\operatorname{Conf}_{n}((\mathbb{C}-\Lambda)/\Lambda), \mathbb{C})\oplus V_{2}\to \mathbb{C}\oplus H^{1}(\operatorname{Conf}_{n}(X\setminus p), \mathbb{C})\oplus V_{1}? $$ Notice that $\mathbb{C}$ represents the $0$-th cohomology group.