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.



Kodaira–Spencer tensor of an isoperiodic deformation
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 $z\_i$


