# Explicit Kodaira-Spencer map of hyperelliptic curves

Let $g\geq 2$, and $$\mathcal T=\{(t_1,\cdots,t_{2g+2})~|~t_i\neq t_j,\forall i\neq j\}.$$ For any $t=(t_1,\cdots,t_{2g+2})\in \mathcal T$, let $$Y_t=\left\{y^2=\prod_{i=1}^{2g+2}(x-t_i)\right\}.$$ Thus we get a family of hyperelliptic curves $\mathcal Y \to \mathcal T$. Let $t^0=(t_1^0,\cdots,t_{2g+2}^0) \in \mathcal T$ be a fixed point (we assume $t_i^0\neq 0,\forall i$). Then we have the Kodaira-Spencer map $$\Theta_{\mathcal T,t^0} \longrightarrow H^1(Y_{t^0}, \Theta_{Y_{t^0}}).$$ Take dual, we get a map (also called Kodiara-Spencer map): $$\kappa:H^0(Y_{t^0}, \omega^{\otimes 2}_{Y_{t^0}}) \longrightarrow \Omega^1_{\mathcal T,t^0}.$$ Now one can take $$\frac{x^i}{y^2}(dx)^2, 0\leq i \leq 2(g-1);~ \frac{x^j}{y}(dx)^2, 0\leq j \leq g-3,$$ as a basis of $H^0(Y_{t^0}, \omega^{\otimes 2}_{Y_{t^0}})$. And take $$dt_1, \cdots, dt_{2g+2},$$ as a basis of $\Omega^1_{\mathcal T,t^0}$. Then can we write down the Kodaira-Spencer map explicitly under the above two basis?

• I don't know the answer off the top of my head, but you might look in some of the classical books on algebraic curves; it's probably treated in some such treatise. One thing does seem clear: You won't need the quadratic differentials that are odd with respect to the hyperelliptic involution $(x,y)\to (x,-y)$; indeed, $\kappa$ will only involve the differentials $x^i(dx/y)^2$, since these are a basis for the cotangent space to the $(2g{-}1)$-dimensional hyperelliptic locus $\mathcal{H}_g$ within the $(3g{-}3)$-dimensional moduli space $\mathcal{M}_g$ of all curves of genus $g$. – Robert Bryant Feb 7 '14 at 20:30
• @Robert Bryant Thank you very much. I don't know any book treating this special problem. Do you have any good reference to this problem? – Pyramid Feb 10 '14 at 11:46
• Already asked (more or less): mathoverflow.net/questions/22345/… – Qfwfq Feb 11 '14 at 18:06
• @Qfwfq: But the answer there is not very explicit... – Robert Bryant Feb 11 '14 at 18:20

Caution: I was thinking over the calculation that led to my proposed answer below, and I realized that I had neglected a term that I haven't actually justified as being zero, so now I'm less sure that these formulae are correct. There may be another term in the formula for $\kappa(\phi_i)$ and I don't see how to rule it out. As a result, I am withdrawing my answer until I have had time to determine whether the extra term really does vanish. My apologies to those who trusted my claim and up-voted it!

Unfortunately, I am not allowed to delete an 'accepted answer', so I'm putting this caution at the top so that readers will know not to regard it as correct (yet).

I haven't checked all of the details, but I think that the following is correct and gives you a basis for an answer: We can get rid of the action of the Möbius group of linear fractional transformations, by, instead, looking at the manifold $$\mathcal{S} = \{(s_0,\ldots,s_{2g-2})\ |\ s_i\not=0,1\ \text{and}\ s_i\not=s_j\ \text{when}\ i\not=j \ \}\subset \mathbb{C}^{2g-1}$$ and letting $s\in\mathcal{S}$ correspond to the genus $g$ hyperelliptic curve $C_s$ defined by $$y^2 = x(x-1)(x-s_0)(x-s_1)\cdots(x-s_{2g-2}).$$ The hyperelliptic involution is $\iota(x,y) = (x,-y)$. A basis for the $\iota$-even holomorphic quadratic differentials on $C_s$ (a space of dimension $2g-1$ when $g>1$) is given by $$\frac{x^i\ dx^2}{x(x-1)(x-s_0)(x-s_1)\cdots(x-s_{2g-2})}\ \text{when}\ 0\le i\le 2g-2,$$ and a basis for the $\iota$-odd holomorphic quadratic differentials on $C_s$ (a space of dimension $g-2$ when $g>1$) is given by $$\psi_j = \frac{x^i\ dx^2}{y}\qquad\text{when}\ 0\le j\le g-3.$$ It turns out that a more convenient basis for the $\iota$-even holomorphic quadratic differentials is $$\phi_i = \frac{dx^2}{x(x-1)(x-s_i)}\qquad \text{when}\ 0\le i\le 2g-2.$$

Then, it seems, in this case, that the (dual) Kodaira-Spencer map is given by $$\kappa(\psi_j) = 0\qquad \text{when}\ 0\le j\le g-3$$ while there is a (nonzero) constant $c$ (which might depend on $g$, but I don't think so) such that $$\kappa(\phi_i) = \frac{c\ ds_i}{s_i(s_i-1)}\qquad \text{when}\ 0\le i\le 2g-2.$$ (This formula even works when $g=1$, by the way.)

I think you can get your desired formula by making this equivariant with respect to the action of the Möbius group $\mathrm{PSL}(2,\mathbb{C})$, but I suspect that the general formula in this case won't be particularly nice.

• Thanks. I wonder how could you get the formula? $$\kappa(\phi_i)=\frac{cds_i}{s_i(s_i-1)}.$$ – Pyramid Feb 12 '14 at 11:10
• My calculation is based on Example 1 (starting at the bottom of page 39 and continuing to the top of page 41) of Complex Manifolds by Morrow and Kodaira (first edition, 1971). I used their ideas as a guide, but I had to change the notation a bit and adapt their arguments. Assuming that I did the calculations correctly (and there was a normalizing constant that I wasn't sure about, hence $c$), the above is the answer that this yields. – Robert Bryant Feb 12 '14 at 12:21