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Kevin H. Lin
Kevin H. Lin
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Let $C$ be a compact Riemann surface, let $C^2$ be the cartesian square of $C$, let $J(C)$ be the degree zero Jacobian of $C$, and let $\delta : C^2 \to J(C)$ be the map $(x,y) \mapsto [\mathcal{O}(x-y)]$.

In this paper http://arxiv.org/abs/math/9810054 of Hain and Reed, page 9, they say that it is an elementary exercise in algebraic topology to show that $\delta^\ast(\phi) = \Delta + (\psi_1 + \psi_2)/2$.

Explanation of notation:

  • $\phi$ denotes the symplectic form $\sum dx_i \wedge dy_i$ on $J(C)$, where $x_i, y_i$ are coordinates on the torus $J(C) = H^1(C;\mathcal{O}) / H^1(C;\mathbb{Z})$ corresponding to a symplectic basis of $H_1(C;\mathbb{Z}) \cong H^1(C;\mathbb{Z})$. This is also the class of the theta divisor $\Theta$.

  • $\psi_i$ is the first Chern class of the relative cotangent bundle of the $i$th projection $C^{2} \to C$.

  • $\Delta$ is the class of the diagonal $C \to C^{2}$, $x \mapsto (x,x)$.

My question is: How to do this "elementary exercise"? It ought to be easy but I'm just not seeing it...

Let $C$ be a compact Riemann surface, let $C^2$ be the cartesian square of $C$, let $J(C)$ be the degree zero Jacobian of $C$, and let $\delta : C^2 \to J(C)$ be the map $(x,y) \mapsto [\mathcal{O}(x-y)]$.

In this paper http://arxiv.org/abs/math/9810054 of Hain and Reed, page 9, they say that it is an elementary exercise in algebraic topology to show that $\delta^\ast(\phi) = \Delta + (\psi_1 + \psi_2)/2$.

Explanation of notation:

  • $\phi$ denotes the symplectic form $\sum dx_i \wedge dy_i$ on $J(C)$, where $x_i, y_i$ are coordinates on the torus $J(C) = H^1(C;\mathcal{O}) / H^1(C;\mathbb{Z})$ corresponding to a symplectic basis of $H_1(C;\mathbb{Z}) \cong H^1(C;\mathbb{Z})$.

  • $\psi_i$ is the first Chern class of the relative cotangent bundle of the $i$th projection $C^{2} \to C$.

  • $\Delta$ is the class of the diagonal $C \to C^{2}$, $x \mapsto (x,x)$.

My question is: How to do this "elementary exercise"? It ought to be easy but I'm just not seeing it...

Let $C$ be a compact Riemann surface, let $C^2$ be the cartesian square of $C$, let $J(C)$ be the degree zero Jacobian of $C$, and let $\delta : C^2 \to J(C)$ be the map $(x,y) \mapsto [\mathcal{O}(x-y)]$.

In this paper http://arxiv.org/abs/math/9810054 of Hain and Reed, page 9, they say that it is an elementary exercise in algebraic topology to show that $\delta^\ast(\phi) = \Delta + (\psi_1 + \psi_2)/2$.

Explanation of notation:

  • $\phi$ denotes the symplectic form $\sum dx_i \wedge dy_i$ on $J(C)$, where $x_i, y_i$ are coordinates on the torus $J(C) = H^1(C;\mathcal{O}) / H^1(C;\mathbb{Z})$ corresponding to a symplectic basis of $H_1(C;\mathbb{Z}) \cong H^1(C;\mathbb{Z})$. This is also the class of the theta divisor $\Theta$.

  • $\psi_i$ is the first Chern class of the relative cotangent bundle of the $i$th projection $C^{2} \to C$.

  • $\Delta$ is the class of the diagonal $C \to C^{2}$, $x \mapsto (x,x)$.

My question is: How to do this "elementary exercise"? It ought to be easy but I'm just not seeing it...

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Kevin H. Lin
Kevin H. Lin
  • 21k
  • 10
  • 116
  • 190

Let $C$ be a compact Riemann surface, let $C^{(2)}$$C^2$ be the symmetriccartesian square of $C$, let $J(C)$ be the degree zero Jacobian of $C$, and let $\delta : C^{(2)} \to J(C)$$\delta : C^2 \to J(C)$ be the map $x+y \mapsto [\mathcal{O}(x-y)]$$(x,y) \mapsto [\mathcal{O}(x-y)]$.

In this paper http://arxiv.org/abs/math/9810054 of Hain and Reed, page 9, they say that it is an elementary exercise in algebraic topology to show that $\delta^\ast(\phi) = \Delta + (\psi_1 + \psi_2)/2$.

Explanation of notation:

  • $\phi$ denotes the symplectic form $\sum dx_i \wedge dy_i$ on $J(C)$, where $x_i, y_i$ are coordinates on the torus $J(C) = H^1(C;\mathcal{O}) / H^1(C;\mathbb{Z})$ corresponding to a symplectic basis of $H_1(C;\mathbb{Z}) \cong H^1(C;\mathbb{Z})$.

  • $\psi_i$ is the first Chern class of the relative cotangent bundle of the $i$th projection $C^{(2)} \to C$$C^{2} \to C$.

  • $\Delta$ is the class of the diagonal $C \to C^{(2)}$$C \to C^{2}$, $x \mapsto x+x$$x \mapsto (x,x)$.

My question is: How to do this "elementary exercise"? I see how to get the $\Delta$ part, because the integral of $\delta^\ast(\phi)$ over the diagonal $\Delta$ is $1$, but I don't see how to get the $\psi$ parts... It ought to be easy but I'm just not seeing it...

Let $C$ be a Riemann surface, let $C^{(2)}$ be the symmetric square of $C$, let $J(C)$ be the degree zero Jacobian of $C$, and let $\delta : C^{(2)} \to J(C)$ be the map $x+y \mapsto [\mathcal{O}(x-y)]$.

In this paper http://arxiv.org/abs/math/9810054 of Hain and Reed, page 9, they say that it is an elementary exercise in algebraic topology to show that $\delta^\ast(\phi) = \Delta + (\psi_1 + \psi_2)/2$.

Explanation of notation:

  • $\phi$ denotes the symplectic form $\sum dx_i \wedge dy_i$ on $J(C)$, where $x_i, y_i$ are coordinates on the torus $J(C) = H^1(C;\mathcal{O}) / H^1(C;\mathbb{Z})$ corresponding to a symplectic basis of $H_1(C;\mathbb{Z}) \cong H^1(C;\mathbb{Z})$.

  • $\psi_i$ is the first Chern class of the relative cotangent bundle of the $i$th projection $C^{(2)} \to C$.

  • $\Delta$ is the class of the diagonal $C \to C^{(2)}$, $x \mapsto x+x$.

My question is: How to do this "elementary exercise"? I see how to get the $\Delta$ part, because the integral of $\delta^\ast(\phi)$ over the diagonal $\Delta$ is $1$, but I don't see how to get the $\psi$ parts... It ought to be easy but I'm just not seeing it...

Let $C$ be a compact Riemann surface, let $C^2$ be the cartesian square of $C$, let $J(C)$ be the degree zero Jacobian of $C$, and let $\delta : C^2 \to J(C)$ be the map $(x,y) \mapsto [\mathcal{O}(x-y)]$.

In this paper http://arxiv.org/abs/math/9810054 of Hain and Reed, page 9, they say that it is an elementary exercise in algebraic topology to show that $\delta^\ast(\phi) = \Delta + (\psi_1 + \psi_2)/2$.

Explanation of notation:

  • $\phi$ denotes the symplectic form $\sum dx_i \wedge dy_i$ on $J(C)$, where $x_i, y_i$ are coordinates on the torus $J(C) = H^1(C;\mathcal{O}) / H^1(C;\mathbb{Z})$ corresponding to a symplectic basis of $H_1(C;\mathbb{Z}) \cong H^1(C;\mathbb{Z})$.

  • $\psi_i$ is the first Chern class of the relative cotangent bundle of the $i$th projection $C^{2} \to C$.

  • $\Delta$ is the class of the diagonal $C \to C^{2}$, $x \mapsto (x,x)$.

My question is: How to do this "elementary exercise"? It ought to be easy but I'm just not seeing it...

Source Link
Kevin H. Lin
Kevin H. Lin
  • 21k
  • 10
  • 116
  • 190

A very basic question about Abel-Jacobi map

Let $C$ be a Riemann surface, let $C^{(2)}$ be the symmetric square of $C$, let $J(C)$ be the degree zero Jacobian of $C$, and let $\delta : C^{(2)} \to J(C)$ be the map $x+y \mapsto [\mathcal{O}(x-y)]$.

In this paper http://arxiv.org/abs/math/9810054 of Hain and Reed, page 9, they say that it is an elementary exercise in algebraic topology to show that $\delta^\ast(\phi) = \Delta + (\psi_1 + \psi_2)/2$.

Explanation of notation:

  • $\phi$ denotes the symplectic form $\sum dx_i \wedge dy_i$ on $J(C)$, where $x_i, y_i$ are coordinates on the torus $J(C) = H^1(C;\mathcal{O}) / H^1(C;\mathbb{Z})$ corresponding to a symplectic basis of $H_1(C;\mathbb{Z}) \cong H^1(C;\mathbb{Z})$.

  • $\psi_i$ is the first Chern class of the relative cotangent bundle of the $i$th projection $C^{(2)} \to C$.

  • $\Delta$ is the class of the diagonal $C \to C^{(2)}$, $x \mapsto x+x$.

My question is: How to do this "elementary exercise"? I see how to get the $\Delta$ part, because the integral of $\delta^\ast(\phi)$ over the diagonal $\Delta$ is $1$, but I don't see how to get the $\psi$ parts... It ought to be easy but I'm just not seeing it...