Kodaira–Spencer map

Remarks

The identification $H^0(X_t, T_t B \otimes \mathcal{O}_{X_t}) = T_t B$ uses that $X_t$ is compact and connected, so its global holomorphic functions are constants. In Čech terms: cover $\mathcal{X}$ near $X_t$ by charts $(z_i, s)$ with $\varpi(z_i, s) = s$, glued by $z_i = f_{ij}(z_j, s)$. For $v = \partial / \partial s$, the class $\rho_t(v)$ is represented by the vector fields $\sum_k \frac{\partial f_{ij}^k}{\partial s}(z_j, t)\, \frac{\partial}{\partial z_i^k}$ on the overlaps.

Example: for a trivial family $X \times B \to B$, the sequence splits, and $\rho_t = 0$ for all $t$.

Example: for the family $\C/(\Z + \tau\Z)$ over the upper half-plane, $\Theta_{X_\tau} \cong \mathcal{O}_{X_\tau}$, so $H^1(X_\tau, \Theta_{X_\tau})$ is one-dimensional, and $\rho_\tau$ is an isomorphism at every $\tau$.

A family with $\rho_t$ surjective at $t$ is complete at $t$: every small deformation of $X_t$ is pulled back from it, by a theorem of Kodaira and Spencer.