Kodaira–Spencer tensor of an isoperiodic deformation

Question

Let $\alpha$ be a holomorphic 1-form on a curve $X$ of genus $g$, which we view as a map of sheaves $\alpha \colon T \to O$. The cokernel of this map is the structure sheaf $O_Z$ of the zero locus $Z \subset X$ of $\alpha$, which is a sum of $2g-2$ skyscraper sheaves (let zeroes $z_i$ of $\alpha$ be simple). It gives rise to the exact sequence $0 \to H^0(O) \to H^0(O_Z) \to H^1(T) \to H^1(O) \to 0$, where the range of the connecting homomorphism is the tangent space of the universal isoperiodic deformation (i. e. the deformations of complex structures on $X$ s. t. the cohomology class $[\alpha] \in H^1(X,\mathbb{C})$ still lies in the corresponding $H^{1,0}$ subspace). On the other hand, the space $H^0(O_Z)/\mathrm{const}$ has natural choice of coordinates, which integrate to the coordinate system on the base of the universal isoperiodic deformation, given by $\left\{\int_{z_0}^{z_i}\alpha\right\}_{i=1}^{2g-3}$ (called in the physics literature 'the relative periods').

My question is the following: how to describe the Kodaira–Spencer tensor associated to a vector from $H^0(O_Z)$? I tried to write down the connecting homomorphism for the Čech complexes, but did not succeed, and did not bother myself again since it must be well-known anyway. However, I did not find a reference.

Answer 1

I suppose you are asking for how to write down pairing of basis elements $v_i\in H^0(O_Z)$ with $\omega\in H^0(\Lambda^1\otimes \Lambda^1)$. Here the simplest answer: locally around $z_i$ let our $\alpha=\alpha(z)dz$ and $\omega=\omega(z)dz^2$ then

$\langle \alpha, \omega\rangle=\oint\limits_{z_i} \frac{\omega dz^2}{\alpha dz}=\oint\limits_{z_i} \frac{\omega}{\alpha}dz$

To see this let's recall that Serre duality is in fact local statement -- it is comming from Yoneda product in $\mathcal{Ext}$'s. We want to write local pairing in Cech terms for covering of the curve with small disks $U_i$ around $z_i$ and complement $U$ to all $z_i$: $$\mathcal{Ext}^0(O,\Lambda^1\otimes \Lambda^1)\otimes \mathcal{Ext}^0(O,O_Z)\xrightarrow{1\otimes\delta}$$

$$\mathcal{Ext}^0(O,\Lambda^1\otimes \Lambda^1)\otimes \mathcal{Ext}^1(\Lambda^1,O)\xrightarrow{\cup} \mathcal{Ext}^1(O,\Lambda^1)$$

The first is given by connecting homomorphism. By inspecting Cech double-complex for $T\to O\to O_Z$ and down-to-earth definition of $\delta$ we obtain that $\delta v_i$ is given by $\frac{1}{\alpha}\partial_z\in T(U_i\cap U)$ as Cech cocyle.

The second map is given by cup-product of Cech cocycles, so it is $\frac{\omega(z)}{\alpha}dz$ on $U_i\cap U$.

Finally we have explicit Cech cocycle as element of $\Lambda^1(U_i\cap U)$ and we need to evaluate $\check{H}^1(\Lambda)\xrightarrow{\int\limits_{C}}\mathbb{C}$. Well, it's given by residue and could be found in general treatment somewhere in Griffits&Harris book. For our curve $C$ it's simple -- take double-complex Cech-Dolbeau resolution of the constant sheaf $\mathbb{C}$, introduce a partition of unity $\rho,\rho_i$ subordinate to $U,U_i$. Then, starting from $a=\Lambda^1(U_i\cap U),\bar{\partial}a=0$ construct by gluing cohomologous element in $\Lambda^{1,1}(C)$ as $W=\bar{\partial}(\rho a)-\bar{\partial}(\rho_i a)\in \Lambda^{1,1}(C)$, by simple complex analysys we obtain that $\int\limits_{C}W= \lim\limits_{\varepsilon\to 0} \int\limits_{C-D_{i,\varepsilon}}W= \lim\limits_{\varepsilon\to 0} \int\limits_{C-D_{i,\varepsilon}}-\bar{\partial}(\rho_i a)= \lim\limits_{\varepsilon\to 0} \oint\limits_{\partial D_{i,\varepsilon}}\rho_i a= 2\pi i\ res_{z_i}(a)$

Here I denote by $D_{i,\varepsilon}$ small disk around $z_i$