Infinitesimal deformation, broadly speaking, considers the infinitesimal variations in the structures of certain mathematical objects. Imagine an object in space, potentially rigid, whose shape changes due to the application of external forces. At each moment in time, $t$, the object assumes a slightly different shape.

In particular, given a compact complex manifold $M$, deformation theory allows us to study the variations of different complex manifold structures on $M$ which depend (either continuously, smoothly, or holomorphically) on a variable $t$. Recall that a compact complex manifold $M$ can be constructed by gluing domains in $\mathbb{C}^n$ via the holomorphic transition functions. By allowing these transition functions to depend on an additional variable $t$, we can create a family of different ways to glue the same domains together. Consequently, we obtain a family of compact complex manifolds $M_t$, each representing a deformation of the original manifold $M$. This concept ties into moduli spaces and moduli problems.

These moduli spaces are central to many areas of algebraic geometry and complex analysis. A moduli space, in broad terms, is a parameter space for a class of objects, meaning that each point in this space represents an equivalence class of these objects. For instance, the moduli space of compact complex manifolds of a given dimension parametrizes all such manifolds up to isomorphism.

When studying deformations of a compact complex manifold $M$, one often examines the tangent space to the moduli space at the point corresponding to $M$. This tangent space can be identified with the first cohomology group of the tangent sheaf of $M$, denoted as $H^1(M, T_M)$. Elements of this cohomology group correspond to infinitesimal deformations of the complex structure on $M$.

As mentioned above, a compact complex manifold $M$ is obtained by gluing domains $U_1,\dots,U_j,\dots, U_l$ in $\Bbb C^n$. We’ll denote the biholomorphic transition maps of the charts $(U_j,z_j)$ by

\[f_{jk} = z_j \circ z^{-1}_k : z_k(U_j \cap U_k) \to z_j(U_j \cap U_k),\]

where $z_j : U_j \to z_j(U_j) \subset \Bbb C^n$ is the map $z_j(p) = (z^{1}_j(p),\dots,z^{n}_j(p))$. Since $z_j(U_j \cap U_k) \subset \Bbb C^n$, $f_{jk}$ can be written in terms of its components as

\[f_{jk} = (f^1_{jk},\dots,f^{n}_{jk})\]

and so $z^\alpha_j = f^\alpha_{jk} \circ z_k$. Without loss of generality, one can assume that the $U_j$’s are polydiscs of radius $1$.

Let $B$ be a domain in $\Bbb C^m$. A complex analytic family of compact complex manifolds is a family of compact complex manifolds $\{M_t \mid t \in B \}$ such that there is a complex manifold $\mathscr{M}$ of dimension $d$ and a holomorphic map $\pi$ from $\mathscr{M}$ onto $B$ satisfying the following conditions:
  1. The rank of the Jacobian of $\pi$ is equal to $m$ at every point of $\mathscr{M}$.
  2. For every $t \in B$, the fiber $\pi^{-1}(t)$ is a compact submanifold of $\mathscr{M}$.
  3. $\pi^{-1}(t) \cong M_t$ for every $t \in B$.

In the above definition, $\mathscr{M}$ is called the total space of the family and $B$ the parameter space. The first condition ensures, utilizing the implicit function theorem, that $\pi^{-1}(t)$ is a closed complex submanifold of $\mathscr{M}$ of dimension $n := d-m$ for each $t \in B$. The second condition just yields that $\pi^{-1}(t)$ are compact complex manifolds and the last condition states that the submanifold $\pi^{-1}(t)$ is biholomorphic to the compact complex manifold $M_t$.

As a set, we have $\mathscr{M} = \bigcup_{t \in B} M_t \times \{t\}$. Now, using the implicit function theorem, one constructs local coordinates for $\mathscr{M}$ such that $\{U_j\}_{j \in J}$ is a locally finite covering and

\[z_j : U_j \to z_j(U_j) \subset \Bbb C^{n+m}\]

is a holomorphic map with $U_j \ni p \mapsto (z^1_j(p),\dots,z^n_j(p),t^1,\dots,t^m)$, where

\[\pi(p) = (t^1,\dots,t^m) = t \in B.\]

Thus each $z_j$ maps $U_j$ into the product $\Bbb C^n \times B \subset \Bbb C^{n+m}$. The manifold $M_t = \pi^{-1}(t)$ has local coordinates given by $(V_j,z_j)$, where $V_j$ belongs to the open cover $\{U_j \cap \pi^{-1}(t)\}$, and $z_j$ is a holomorphic map on $V_j \cap \pi^{-1}(t)$ given by $p \mapsto z_j(p)=(z^1_j(p),\dots,z^n_j(p))$.

The transition functions $f_{jk} = z_j \circ z^{-1}_k$ will now also incorporate the parameter $t$ from where we see that the compact complex manifold $M_{t_0}$ for $t_0 \in B$ is obtained by gluing finite number of open sets

\[z_j(U_j)_{t_0} = \{\pi^{-1}(t_0) \cap z_j(U_j) \mid j \in J\}\]

of $\Bbb C^n$ via $f_{jk}$.

Let $M$ and $N$ be two compact complex manifolds and $B$ a domain in $\Bbb C^n$. $N$ is called a deformation of $M$ if there exists an analytic family $(\mathscr{M}, B, \pi)$ such that $M = \pi^{-1}(t_0)$ and $N = \pi^{-1}(t_1)$, for some $t_0,t_1 \in B$.

For the notion of equivalence between complex analytic families $(\mathscr{M}, B, \pi_\mathscr{M})$ and $(\mathscr{N}, B, \pi_\mathscr{N})$ we have the following definition.

Let $(\mathscr{M}, B, \pi_\mathscr{M})$ and $(\mathscr{N}, B, \pi_\mathscr{N})$ be complex analytic families. Then $(\mathscr{M}, B, \pi_\mathscr{M})$ and $(\mathscr{N}, B, \pi_\mathscr{N})$ are equivalent if there exists a biholomorphic map $F : \mathscr{M} \to \mathscr{N}$ such that for each $t \in B$, $F$ maps $M_t = \pi^{-1}_\mathscr{M}(t)$ biholomorphically to $N_t = \pi^{-1}_\mathscr{N}(t)$.

Considering the simplest possible choice for $\mathscr{M}$, we give a notion of triviality for these complex analytic families.

Let $(\mathscr{M}, B, \pi)$ be a complex analytic family and $M = \pi^{-1}(t_0)$ for a fixed $t_0 \in B$. Then $(\mathscr{M}, B, \pi)$ is called trivial if it is equivalent to $(M \times B, B, \pi)$.

As you might expect, we also have a notion of local triviality. Before giving you the definition, note that if $I \subset B$ is a subdomain of $B$, then $(\mathscr{M}_I, I, \pi)$ yields a complex analytic family with $\mathscr{M}_I = \pi^{-1}(I)$.

A complex analytic family $(\mathscr{M}, B, \pi)$ is said to be locally trivial if, for each $t \in B$, there is a subdomain $I \subset B$ with $t \in I$ such that $(\mathscr{M}_I, I, \pi)$ is trivial.

If $(\mathscr{M}, B, \pi)$ is a complex analytic family and $f : D \to B$ a holomorphic map of complex manifolds, one can ask can we produce a complex analytic family with $D$ as the base space? In other words, can we perform a “change of parameters”? The answer to this question is affirmative and the way to proceed is by first considering the pullback $\mathscr{M} \times_B D$ of the diagram

\[\xymatrix{ & \mathscr{M} \ar@{->}[d]^{\pi} \\ D \ar@{->}[r]_{h} & B }\]

and then consider the image of $\mathscr{M} \times_B D$ under the map

\[\Pi : \mathscr{M} \times D \to B \times D.\]

This yields the graph $G_h$ of $h$, which is a submanifold of $B \times D$ and is biholomorphic to $D$ via $p : B \times D \to D$. Since $\pi$ is a holomorphic map with maximal rank, the map $\pi \times \operatorname{id}_D$ is also. It follows that $\mathscr{M} \times_B D = (\pi \times \operatorname{id}_D)^{-1}(G_h)$ obtains a complex submanifold structure of $\mathscr{M} \times B$. The complex analytic family $(\mathscr{M} \times_B D, D, p \circ \Pi)$ is called the induced family or pullback from $(\mathscr{M}, B, \pi)$ via $h$.

The next thing on our agenda is to consider, given a compact complex manifold $M$ that occurs as the fiber over some $t_0 \in B$ of a complex analytic family $(\mathscr{M}, B, \pi)$, for which $t \in B$ the manifold $M_t = \pi^{-1}(t)$ is a deformation of $M$. The following theorem provides an answer to this.

Let $(\mathscr{M}, B, \pi)$ be a complex analytic family of compact complex manifolds and $M_{t_0} = \pi^{-1}(t_0)$ a compact complex manifold for a fixed $t_0 \in B$, obtained by gluing polydiscs $\{U_j\}_{j \in J}$ via the transition maps $f_{jk}$ defined on $U_j \cap U_k \ne \emptyset$. Then there exists a domain $I \subset B$ such that for each $t \in I$, $M_t = \pi^{-1}(t)$ is a compact complex manifold obtained by gluing the same polydiscs $\{U_j\}_{j \in J}$ using different transition maps $f_{jk}(z_k, t)$, with the initial condition $f_{jk}(z_k,t_0) = f_{jk}(z_k)$.
Consider a locally finite open cover $\{U_j\}_{j \in J}$ for $\mathscr{M}$ and local coordinates $\{(U_j,z_j)\}_{j \in J}$ with $z_j(U_j) = V_j \times I_j$, where $V_j$ can be assumed to be a polydisc in $\Bbb C^n$ and $I_j \subset B$ a polydisc in $\Bbb C^m$. Let $\{f_{jk}\}$ be the transition maps that glue the domains $\{V_j \times I_j\}$ to form $\mathscr{M}$. For $t_0 \in B$, $M_{t_0} = \pi^{-1}(t_0)$ is the compact complex manifold obtained by gluing $$ \{V_j \times \{t_0\} \cong V_j \mid U_j \cap M_{t_0} \ne \emptyset \} $$ via the transition maps $\{f_{jk}(z_k,t_0)\}$. Since the fibers of $\pi$ are compact and the cover is locally finite, there exists an open neighborhood $I$ of $t_0 \in B$ that can be identified with a polydisc in $\Bbb C^m$ such that $\pi^{-1}(I)$ is contained in the union of a finite number of $U_j$'s. Set $U'_j = U_j \cap M_{t_0}$. Then by construction, for any $t \in I$ the compact complex manifold $\pi^{-1}(t) = M_t$ is obtained by gluing $\{ V_j \times \{t\} \cong V_j \mid j = 1,\dots,l\}$ via the transition maps $\{f_{jk}(z_k,t)\}$. Hence, the restriction $(\mathscr{M}_I, I, \pi)$ of $(\mathscr{M}, B, \pi)$ to $I$ consists of fibers which are all obtained by gluing the same polydiscs $\{V_1,\dots,V_l\}$ with the transition maps $f_{jk}(z_k,t)$ depending on $t \in I$.

We’ll now turn to formalize the thing we’ve been calling “infinitesimal deformation”. For a complex analytic family $(\mathscr{M}, B, \pi)$, we seek to give a meaningful definition for

\[\dfrac{\partial}{\partial t}M_t.\]

Meaningful in the sense that $\frac{\partial}{\partial t}M_t \Bigr|_{t_0} = 0$ if $(\mathscr{M}, B, \pi)$ is locally trivial at $t_0 \in B$. Since a deformation of a compact complex manifold $M$ with a cover ${U_j}$ is given by different gluings with the transition maps ${f_{jk}}$, one would expect the infinitesimal deformation to be determined by the datum of $\left\{\frac{\partial}{\partial t} f_{jk}\right\}$.

Let’s begin by looking at what this data gives us with a simple example. Consider $\mathscr{M} = \bigcup_{t \in \Bbb C^\ast} \Bbb CP^1 \times \{t\}$ with $\Bbb CP^1 = U_0 \cup U_1$ being covered by $U_0 = \{[w_0:w_1] \mid w_0 \ne 0\}$ and $U_1 = \{[w_0:w_1] \mid w_1 \ne 0\}$. We have two chart maps given by

\[\begin{align*} z_0 : U_0 &\to \Bbb C \\ [w_0 : w_1] &\mapsto t\frac{w_1}{w_0} \end{align*}\]

and

\[\begin{align*} z_1 : U_1 &\to \Bbb C \\ [w_0 : w_1] &\mapsto t\frac{w_0}{w_1}. \end{align*}\]

Now $z^{-1}_0(\lambda) = [1 : \lambda/t]$ and likewise for $z^{-1}_1$. The transition functions are thus given by

\[f_{01}(z_1,t) = \frac{t^2}{z_1}\]

and

\[f_{10}(z_0,t) = \frac{t^2}{z_0}.\]

Thus $\frac{\partial f_{01}}{\partial t} = \frac{2t}{z_1} \ne 0$ and $\frac{\partial f_{10}}{\partial t} = \frac{2t}{z_0} \ne 0$. However the coordinates $z_0$ and $z_1$ are clearly compatible with the standard coordinates $z’_0 = \frac{w_1}{w_0}$ and $z’_1 = \frac{w_0}{w_1}$ so $\Bbb CP^1_t \cong \Bbb CP^1$.

The moral here is that we need something stronger than $\left\{\frac{\partial}{\partial t} f_{jk}\right\}$. Note that the transition maps satisfy

\[f_{ik}(z_k,t) = f_{ij}(f_{jk}(z_k,t),t)\]

on $U_i \cap U_j \cap U_k \ne \emptyset$. In more detail we have

\[f_{ik}^{\alpha }(z_{k},t)=f_{ij}^{\alpha }(f_{jk}^{1}(z_{k},t),\ldots ,f_{jk}^{n}(z_{k},t),t),\]

for $\alpha = 1,\dots,n$. Now $z^\alpha_j = f^{\alpha}_{jk}(z_k,t)$ so from

\[{\displaystyle {\begin{aligned}{\frac {\partial f_{ik}^{\alpha }(z_{k},t)}{\partial t}}&={\frac {\partial f_{ij}^{\alpha }(z_{j},t)}{\partial t}}+\sum _{\beta =0}^{n}{\frac {\partial f_{ij}^{\alpha }(z_{j},t)}{\partial f_{jk}^{\beta }(z_{k},t)}}\cdot {\frac {\partial f_{jk}^{\beta }(z_{k},t)}{\partial t}}\\\end{aligned}}}\]

we obtain

\[{\displaystyle {\begin{aligned}{\frac {\partial f_{ik}^{\alpha }(z_{k},t)}{\partial t}}&={\frac {\partial f_{ij}^{\alpha }(z_{j},t)}{\partial t}}+\sum _{\beta =0}^{n}{\frac {\partial z_{i}^{\alpha }}{\partial z_{j}^{\beta }}}\cdot {\frac {\partial f_{jk}^{\beta }(z_{k},t)}{\partial t}}\\\end{aligned}}}.\]

Also

\[{\displaystyle {\frac {\partial }{\partial z_{j}^{\beta }}}=\sum _{\alpha =1}^{n}{\frac {\partial z_{i}^{\alpha }}{\partial z_{j}^{\beta }}}\cdot {\frac {\partial }{\partial z_{i}^{\alpha }}}}\]

and so we obtain the following

\[{\displaystyle {\begin{aligned}\sum _{\alpha =0}^{n}{\frac {\partial f_{ik}^{\alpha }(z_{k},t)}{\partial t}}{\frac {\partial }{\partial z_{i}^{\alpha }}}=&\sum _{\alpha =0}^{n}{\frac {\partial f_{ij}^{\alpha }(z_{j},t)}{\partial t}}{\frac {\partial }{\partial z_{i}^{\alpha }}}\\&+\sum _{\beta =0}^{n}{\frac {\partial f_{jk}^{\beta }(z_{k},t)}{\partial t}}{\frac {\partial }{\partial z_{j}^{\beta }}.}\\\end{aligned}}}\]

Then defining

\[\theta_{ik}(t) = \sum _{\alpha =0}^{n}\frac {\partial f_{ik}^{\alpha }(z_{k},t)}{\partial t}\frac {\partial }{\partial z_{i}^{\alpha }},\]

the equality becomes

\[\theta_{ik}(t) = \theta_{ij}(t) + \theta_{jk}(t).\]

Note that $\theta_{jk}(t) - \theta_{ik}(t) + \theta_{ij}(t) = 0$ and for $i = k$ since $f^\alpha_{kk} = z^\alpha_k$, we have that $\theta_{kk}(t) = 0$ as well as $\theta_{kj}(t) = -\theta_{jk}(t)$. It follows that ${\theta_{jk}(t)}$ forms a $1$-cocycle. Let $T_{M_t}$ be the sheaf of holomorphic vector fields on $M_t$, then

\[\theta_{jk}(t) \in \Gamma(U_j \cap U_k, T_{M_t}).\]

So ${\theta_{jk}(t)} \in Z^1(\mathcal{U}_t, T_{M_t})$, where $\mathcal{U}_t = {U_j}$ is the covering $M_t = \bigcup U_j$. We denote the corresponding cohomology class $\theta(t) \in H^1(\mathcal{U}_t,T_{M_t}) \subset H^1(M_t,T_{M_t})$. We call $\theta(t)$ the infinitesimal deformation of $M_t$ and denote it by

\[\frac{\partial}{\partial t} M_t := \theta(t).\]

There are a few things we need to check here. The first question arising is that is $\frac{\partial}{\partial t} M_t$ independent of the choice of local data?

The infinitesimal deformation $\frac{\partial}{\partial t} M_t$ independent of the choice of local coordinates.
First, fix a complex analytic family $(\mathscr{M}, B, \pi)$ with of local coordinates $\left(U_{i}, z_{i}\right)_{i \in I}$, we show that $\theta(t)$ does not change if we consider a refinement of the open covering $\mathcal{U}=\left\{U_{i}\right\}_{i \in I}$ of $\mathscr{M}$. Let $\mathcal{V}=\left\{V_{j}\right\}_{j \in J}$ be any refinement of $\mathcal{U}=\left\{U_{i}\right\}_{i \in I}$, such that $V_{j} \subset U_{i(j)}$ for all $j \in J$. Let $z_{j}^{\prime}: V_{j} \longrightarrow \mathbb{C}^{n}$ be a local coordinate defined on $V_{j}$ for each $j \in J$, which is the restriction of the local coordinate $z_{i(j)}: U_{i(j)} \longrightarrow \mathbb{C}^{n}$. Thus the holomorphic vector field $\theta_{j k}^{\prime}(t)$ is the restriction of $\theta_{i(j) i(k)}(t)$ to $V_{j} \cap V_{k} \cap M_t$. In fact, setting $\mathcal{B}_{t}=\left\{V_{j t}\right\}_{j \in J}$ where $V_{j t}=V_{j} \cap M_t \neq \emptyset$, we have $$ \theta_{j k}^{\prime}(t)=\theta_{i(j) i(k)}^{\prime}(t)_{\mid V}, \quad V=V_{j t} \cap V_{k t} $$ Hence the 1-cocycles $\left\{\theta_{j k}(t)\right\} \in Z^{1}(\mathcal{U}, T_{M_t})$ and $\left\{\theta_{j k}^{\prime}(t)\right\} \in Z^{1}(\mathcal{V}, T_{M_t})$ satisfy $$ \left\{\theta_{j k}^{\prime}(t)\right\}=\Pi_{\mathfrak{B}_{t}}^{\mathcal{U}_{t}}\left(\left\{\theta_{j k}(t)\right\}\right). $$ Now, we consider two sets of local coordinates $\left(V_{l}, z_{l}\right)_{l \in L}$ and $\left(V_{\lambda}^{\prime}, z_{\lambda}^{\prime}\right)_{\lambda \in \Lambda}$. Since two locally finite coverings have a common refinement and since $\theta(t)$ does not depend on the choice of refinement, we can work with local coordinates $\left(U_{i}, z_{i}\right)_{i \in I}$ consisting of open subsets belonging to a refinement of both $\left\{V_{l}\right\}_{l \in L}$ and $\left\{V_{\lambda}^{\prime}\right\}_{l \in \Lambda}$, as well as the restrictions of the local coordinates $z_{l}$ and $z_{\lambda}^{\prime}$ to those open subsets. Therefore given $z_{i}$ and $w_{i}$ defined on each $U_{i}$, we want to show that the infinitesimal deformation $\eta(t)$ defined with respect to $w_{i}$ coincides with $\theta(t)$. That is, on intersections, $\{\theta_{jk}(t)\} -\{\eta_{jk}(t)\}$ is a coboundary. If $h_{j k}: w_{k} \longmapsto w_{j}$ are transition maps for the local coordinates $\left\{w_{i}\right\}$, then we obtain the holomorphic vector field $$ \begin{equation*} \eta_{j k}(t)=\sum_{\beta=1}^{n} \frac{\partial h_{j k}^{\beta}\left(w_{k}, t\right)}{\partial t} \frac{\partial}{\partial w_{j}^{\beta}}, \quad w_{k}=h_{k j}\left(w_{j}, t\right) \end{equation*} $$ Consider now the cohomology class $\eta(t) \in H^1(M_t, T_{M_t})$ corresponding to the cocycle $\{\eta_{jk}(t)\} \in Z^1(\mathcal{U}_t, T_{M_t})$. For a holomorphic function $g^\alpha_j$ on the variables $z^1_j,\dots,z^n_j,t$ we have that $w^\alpha_j = g^\alpha_j(z_j,t)$. Since $$ g_{j}^{\alpha}\left(z_{j}, t\right)=w_{j}^{\alpha}=h_{j k}^{\alpha}\left(w_{k}, t\right)=h_{j k}^{\alpha}\left(g_{k}\left(z_{k}, t\right), t\right) $$ and $z_{j}=f_{j k}\left(z_{k}, t\right)$, we obtain $$ \begin{equation*} g_{j}^{\alpha}\left(f_{j k}\left(z_{k}, t\right), t\right)=h_{j k}^{\alpha}\left(g_{k}\left(z_{k}, t\right), t\right) \end{equation*} $$ on $U_{j} \cap U_{k} \neq \emptyset$. Differentiating this with respect to $t$ yields $$ \sum_{\beta=1}^{n} \frac{\partial g_{j}^{\alpha}}{\partial z_{j}^{\beta}} \frac{\partial f_{j k}^{\beta}}{\partial t}+\frac{\partial g_{j}^{\alpha}}{\partial t}=\sum_{\beta=1}^{n} \frac{\partial h_{j k}^{\alpha}}{\partial w_{k}^{\beta}} \frac{\partial g_{k}^{\beta}}{\partial t}+\frac{\partial h_{j k}^{\alpha}}{\partial t} $$ i.e., $$ \sum_{\beta=1}^{n} \frac{\partial g_{j}^{\alpha}}{\partial t}+\frac{\partial w_{j}^{\alpha}}{\partial z_{j}^{\beta}} \frac{\partial f_{j k}^{\beta}}{\partial t}=\sum_{\beta=1}^{n} \frac{\partial w_{j}^{\alpha}}{\partial w_{k}^{\beta}} \frac{\partial g_{k}^{\beta}}{\partial t}+\frac{\partial h_{j k}^{\alpha}}{\partial t}. $$ Multiplying by $\frac{\partial}{\partial w_{j}^{\alpha}}$ from the right and taking the summation $\sum_{\alpha=1}^{n}$, we furthermore obtain the following equality $$ \sum_{\beta=1}^{n} \frac{\partial f_{j k}^{\beta}}{\partial t} \frac{\partial}{\partial z_{j}^{\beta}}+\sum_{\alpha=1}^{n} \frac{\partial g_{j}^{\alpha}}{\partial t} \frac{\partial}{\partial w_{j}^{\alpha}}=\sum_{\beta=1}^{n} \frac{\partial g_{k}^{\beta}}{\partial t} \frac{\partial}{\partial w_{k}^{\beta}}+\sum_{\alpha=1}^{n} \frac{\partial h_{j k}^{\alpha}}{\partial t} \frac{\partial}{\partial w_{j}^{\alpha}} $$ where we use the equalities $$ \frac{\partial}{\partial z_{j}^{\beta}}=\sum_{\alpha=1}^{n} \frac{\partial w_{j}^{\alpha}}{\partial z_{j}^{\beta}} \frac{\partial}{\partial w_{j}^{\alpha}} $$ and $$ \frac{\partial}{\partial w_{k}^{\beta}}=\sum_{\alpha=1}^{n} \frac{\partial w_{j}^{\alpha}}{\partial w_{k}^{\beta}} \frac{\partial}{\partial w_{j}^{\alpha}}. $$ Now setting $\theta_{j}(t)=\sum_{\alpha=1}^{n} \frac{\partial g_{j}^{\alpha}\left(z_{j}, t\right)}{\partial t} \frac{\partial}{\partial w_{j}^{\alpha}}, w_{j}^{\alpha}=g_{j}^{\alpha}\left(z_{j}, t\right)$, we have $$ \begin{equation*} \theta_{j k}(t)-\eta_{j k}(t)=\theta_{k}(t)-\theta_{j}(t). \end{equation*} $$ Since $\theta_{j}(t)$ is a holomorphic vector field on $U_{j t}=U_{j} \cap M_{t}$, there exists a coboundary $\left\{\theta_{j}(t)\right\} \in C^{0}\left(\mathcal{U}_{t}, \Theta_{t}\right)$, moreover we obtain $$ \left\{\theta_{j k}(t)\right\}-\left\{\eta_{j k}(t)\right\}=\delta\left(\left\{\theta_{j}(t)\right\}\right), $$ where $\left\{\theta_{j k}(t)\right\} \in Z^{1}\left(\mathcal{U}_{t}, \Theta_{t}\right)$. Therefore $\eta(t)$ coincides with $\theta(t)$, and the infinitesimal deformation $\frac{\partial}{\partial t} M_t$ does not depend on the choice of local coordinates.
If $(\mathscr{M}, B, \pi)$ is locally trivial, then $$ \frac{\partial}{\partial t} M_t = 0. $$
Suppose that $(\mathscr{M}, B, \pi)$ is locally trivial. Then there exists an open set $I \subset B$ such that $(\mathscr{M}_I, I, \pi)$ is equivalent to $(M \times B, B, \pi)$, where $M = \pi^{-1}(t_0)$ for a fixed $t_0 \in B$. Let $\{w_j\}_{j \in J}$ be local coordinates of $M$. since $\frac{\partial}{\partial t} M_t$ does not depend on the chosen local coordinates, we can construct $\theta(t)$ for $t \in I$ in terms of the local coordinates $\{u_j\}$ given by $u_j = (w_j,t)$ of $M \times I$. Let $h_{jk}$ be the transition map $$ (w_j, t) \mapsto (w_k,t) = (h_{jk}(w_k),t). $$ Since $h_{jk}$ is independent of $t$, we have $$ \theta_{jk}(t) = \sum_{\beta=1}^n \frac{\partial h^\beta_{jk}(w_k)}{\partial t} \frac{\partial}{\partial u^\beta_j} = 0. $$ Thus $\frac{\partial}{\partial t} M_t = 0$.

Referring back to our example on the projective space $\mathscr{M} = \bigcup_{t \in \Bbb C^\ast} \Bbb CP^1 \times \{t\}$, we obtained $\frac{\partial f_{01}}{\partial t} = \frac{2t}{z_1}$. This gives

\[\theta_{01}(t) = \frac{2t}{z_1}\frac{\partial}{\partial z_0}.\]

Note that $\frac{\partial z_0}{\partial z_1} = -\frac{t^2}{z^2_1}$ since on $U_0 \cap U_1$ we have $z_0 = \frac{t^2}{z_1}$. Also

\[\frac{\partial}{\partial z_0} = \frac{\partial z_1}{\partial z_0}\frac{\partial}{\partial z_1} = -\frac{z^2_1}{t^2}\frac{\partial}{\partial z_1}\]

on the intersection. Hence

\[\begin{align*} \theta_{01}(t) &= 2 \frac{z_0}{t}\frac{\partial}{\partial z_0} \\ &= - 2 \frac{z_1}{t}\frac{\partial}{\partial z_1} \\ &= \frac{z_0}{t}\frac{\partial}{\partial z_0} - \frac{z_1}{t}\frac{\partial}{\partial z_1}, \end{align*}\]

and so $\theta_{01}(t)$ is a coboundary $\theta_{01}(t) = \theta_0(t) - \theta_1(t)$ where $\theta_i(t) = \frac{z_i}{t}\frac{\partial}{\partial z_i}$. The converse of the previous proposition is not in general true without forcing an additional condition on the dimension of $H^1(M_t, T_{M_t})$.

Let $(\mathscr{M}, B, \pi)$ be a complex analytic family and $M = \pi^{-1}(t_0)$ for some $t_0 \in B$. If $\dim H^1(M_t, T_{M_t})$ does not depend on $t$ and $$ \frac{\partial}{\partial t}M_t \equiv 0, $$ then $(\mathscr{M}, B, \pi)$ is locally trivial.

To sum up the above discussions, I’ll now give you the definition of the famous Kodaira-Spencer map that one encounters more often than not when working on deformation theory.

The $\Bbb C$-linear map $$ \begin{align*} \rho_t : T_t B &\to H^1(M_t, T_{M_t}) \\ \frac{\partial}{\partial t} &\mapsto \frac{\partial}{\partial t}M_t \end{align*} $$ is called the Kodaira-Spencer map at $t \in B$ for the complex analytic family $(\mathscr{M}, B, \pi)$.

For a slightly different viewpoint, let $X$ be a compact analytic space (or a complete scheme). A deformation of $X$ parameterized by a pointed analytic space (pointed scheme) $(Y,y_0)$ is a proper flat morphism

\[\varphi : \mathscr{X} \to (Y, y_0)\]

plus a given isomorphism between $X$ and the fiber $\varphi^{-1}(y_0)$. A morphism of deformations between $\varphi : \mathscr{X} \to (Y,y_0)$ and $\varphi’ : \mathscr{X}’ \to (Y’,y’_0)$ is a cartesian diagram

\[\xymatrix{ \mathscr{X} \ar@{->}[d]_{\varphi} \ar@{->}[r]^{\alpha} & \mathscr{X}' \ar@{->}[d]^{\varphi'} \\ (Y,y_0) \ar@{->}[r]^{\beta} & (Y',y'_0) }\]

where $\alpha$ and $\beta$ are morphisms inducing the identity on $X$. Now an infinitesimal deformation of $X$ is simply a deformation of $X$ parameterized by $\operatorname{Spec}\Bbb C[\varepsilon]$. Again, $X$ is given by the transition data $\{U_i,z_i,f_{ij}(z_j)\}$. The total space $\mathscr{X}$ of an infinitesimal deformation $\varphi : \mathscr{X} \to \operatorname{Spec}\Bbb C[\varepsilon]$ of $X$ may be thought of as being given by gluing $U_i \times \operatorname{Spec}\Bbb C[\varepsilon]$ via the identifications

\[z_i = \widetilde{f}_{ij}(z_j,\varepsilon) = f_{ij}(z_j) + \varepsilon b_{ij}(z_j),\]

while $\varphi$ is given locally by $\varphi(z_i,\varepsilon) = \varepsilon$. These $\widetilde{f}_{ij}$ also have to satisfy the cocycle rule

\[\widetilde{f}_{ij}(\widetilde{f}_{jk}(z_k,\varepsilon),\varepsilon) = \widetilde{f}_{ik}(z_k,\varepsilon),\]

which gives

\[\widetilde{f}_{ik}(z_k,\varepsilon) = f_{ij}(f_{jk}(z_k) + \varepsilon b_{jk}(z_k)) + \varepsilon b_{ij}(f_{jk}(z_k) + \varepsilon b_{jk}(z_k)).\]

Since the dual numbers satisfy $g(a+b\varepsilon )=g(a)+\varepsilon g’(a)b$ we obtain

\[f_{ij}(f_{jk}(z_k) + \varepsilon b_{jk}(z_k)) = f_{ij}(f_{jk}(z_k)) + \varepsilon \frac{\partial f_{ij}(z_i)}{\partial z_i}b_{jk}(z_k)\]

and

\[\begin{align*} \varepsilon b_{ij}(f_{jk}(z_k) + \varepsilon b_{jk}(z_k)) &= \varepsilon b_{ij}(f_{jk}(z_k)) + \varepsilon^2\frac{\partial b_{ij}(z_i)}{\partial z_i}b_{jk}(z_k)\\ &= \varepsilon b_{ij}(f_{jk}(z_k)) \\ &= \varepsilon b_{ij}(z_j). \end{align*}\]

Hence the cocycle condition on $\widetilde{f}_{ij}$ reduces to the cocycle condition on $f_{ij}$ plus $b_{ik} = \frac{\partial f_{ij}}{\partial z_j}b_{jk} + b_{ij}$. One can then form $\theta_{ij} = \sum_{\alpha = 1}^n b^\alpha_{ij}\frac{\partial}{\partial z^\alpha_i}$ and see that the $1$-cochain $\theta = {\theta_{ij}} \in C^1(\mathcal{U}, T_X)$ is a cocycle. The class

\[[\theta] \in H^1(X, T_X)\]

defines the Kodaira-Spencer class. Note that there is an exact sequence of sheaves of $\mathcal{O}_X$-modules

\[0 \longrightarrow T_X \longrightarrow T_{\mathscr{X}} \longrightarrow \varphi^\ast T_{\operatorname{Spec}\Bbb C[\varepsilon]} \longrightarrow 0.\]

Passing this to the cohomology sequence one obtains the Kodaira-Spencer class as the coboundary of

\[\varphi^\ast\left(\frac{\partial}{\partial \varepsilon}\right) \in H^0(X, T_{\operatorname{Spec}\Bbb C[\varepsilon]} ).\]

Hence $[\theta]$ depends only on the equivalence class of the first-order deformation $\mathscr{X} \to \operatorname{Spec}\Bbb C[\varepsilon]$.

Much more can be said about the theory, but I have other things that I need to focus on for now so we’ll leave this here for now and maybe come back to it later on.