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...



A very basic question about Abel-Jacobi map
Let me write $x\_i$ and $y\_i$ for a symplectic basis of cohomology of $C$, and $a\_i$, $b\_i$ for the linear dual basis of the first homology of $C$. It is enough to find $\delta^\*(dx\_i)$ and $\delta^\*(dy\_i)$ in terms of $x\_i$ and $y\_i$.


But to do this we just evaluate $\langle \delta^\*(dx\_i), a\_i \otimes 1 \rangle$, $\langle \delta^\*(dx\_i), b\_i \otimes 1 \rangle$ and so on, which is easy as $\delta\_\*(a\_i \otimes 1) = a\_i$, $\delta\_\*(1 \otimes a\_i) = -a\_i$ and so on.


Hence $\delta^\*(dx\_i) = x\_i \otimes 1 - 1 \otimes x\_i$ and $\delta^\*(dy\_i) = y\_i \otimes 1 - 1 \otimes y\_i$. Hence
$$\delta^\*(\phi) = \sum (x\_i \otimes 1 - 1 \otimes x\_i)(y\_i \otimes 1 - 1 \otimes y\_i)$$
and multiplying the whole mess out and using relations like $x\_iy\_i = [C] = \tfrac{1}{\chi(C)} \psi$ (which perhaps has a sign, depending on your conventions), you get the required expression.




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Addendum: You may also be interested in the following. In "[Relations among tautological classes revisited](http://arxiv.org/PS_cache/arxiv/pdf/1012/1012.1430v1.pdf)" I gave the following alternative description of the class $\delta^\*(\phi)$ on $\mathcal{C}\_g^2$, the square of the universal curve, which parametrises curves with two ordered non-necessarily distinct points. The paper uses different notation, but I will stick to yours.


Let $\pi : \overline{\mathcal{C}}\_g^2 \to \mathcal{C}\_g^2$ be the universal curve, and $\Delta\_i \in H^2(\overline{\mathcal{C}}\_g^2)$ for $i=1,2$ denote the class of the locus of the first and second points respectively. Then
$$\delta^\*(\phi) = -\pi\_!((\Delta\_1 - \Delta\_2)^2).$$
Again, depending on your conventions there may be a sign and a power of 2 somewhere.


