If $\mathcal{X} = X \times B \to B$ is the constant (trivial) family, then $\mathrm{KS}_b = 0$ for all $b$. The classifying map $B \to \mathcal{M}$ is constant, and its differential is zero.

If $\mathcal{X} \to \mathcal{M}$ is the universal family (when it exists), then $\mathrm{KS}_{[X]}: T_{[X]}\mathcal{M} \to H^1(X, T_X)$ is an isomorphism at every point. This is the tautological statement that the tangent space to moduli equals the space of first-order deformations.

Consider the Legendre family $E_\lambda: y^2 = x(x-1)(x-\lambda)$ over $B = \mathbb{A}^1 \setminus \lbrace 0, 1 \rbrace$. The KS map at $\lambda_0$ is: $\mathrm{KS}_{\lambda_0}: T_{\lambda_0} B = k \to H^1(E_{\lambda_0}, T_{E_{\lambda_0}})$ Since $H^1(E, T_E) \cong H^0(E, \omega_E)^{\vee} \cong k$ (one-dimensional), the KS map is either zero or an isomorphism. It is an isomorphism because the $j$-invariant $j(\lambda) = 256 \frac{(\lambda^2 - \lambda + 1)^3}{\lambda^2(\lambda - 1)^2}$ has nonvanishing derivative for generic $\lambda$.

In coordinates, the class $\mathrm{KS}(\partial/\partial\lambda)$ can be represented by the Cech cocycle obtained by differentiating the transition functions of the family with respect to $\lambda$.

Let $f: \mathcal{C} \to B$ be a family of smooth curves of genus $g \geq 2$. Then: $\mathrm{KS}_b: T_b B \to H^1(C_b, T_{C_b}) \cong H^0(C_b, \omega_{C_b}^{\otimes 2})^{\vee}$ The dual map is: $\mathrm{KS}_b^{\vee}: H^0(C_b, \omega_{C_b}^{\otimes 2}) \to T_b^* B$ This sends a quadratic differential $q \in H^0(C_b, \omega_C^2)$ to a cotangent vector on the base. In Teichmuller theory, quadratic differentials on a Riemann surface parametrize the cotangent space to Teichmuller space.

Consider the family of genus-$g$ hyperelliptic curves $C_t: y^2 = \prod_{i=1}^{2g+2} (x - t_i)$ over $B = \lbrace (t_1, \ldots, t_{2g+2}) \mid t_i \text{ distinct} \rbrace / \mathrm{PGL}_2$. The dimension of $B$ is $2g + 2 - 3 = 2g - 1$, while $\dim H^1(C, T_C) = 3g - 3$. So: $\mathrm{KS}: k^{2g-1} \to k^{3g-3}$ is injective for $g \geq 2$ (the family captures only the hyperelliptic deformations, missing $g - 2$ non-hyperelliptic directions). For $g = 2$, it is an isomorphism (all genus-2 curves are hyperelliptic).

For a family of line bundles $\mathcal{L}_t$ on a curve $C$ parametrized by $t \in B$, the KS map is: $\mathrm{KS}: T_b B \to H^1(C, \mathcal{O}_C) = T_{[\mathcal{L}_b]} \mathrm{Pic}^0(C)$ If $B = C$ and $\mathcal{L}_t = \mathcal{O}_C(t - p_0)$ (varying the point), this is the differential of the Abel-Jacobi map $\mathrm{AJ}: C \to \mathrm{Pic}^1(C) \cong J(C)$: $d(\mathrm{AJ})_p: T_p C \to T_{[\mathcal{O}(p-p_0)]} J(C) = H^1(C, \mathcal{O}_C) \cong H^0(C, \omega_C)^{\vee}$ The dual of this map sends $\omega \in H^0(C, \omega_C)$ to $\omega(p)$, the evaluation at $p$. This is an embedding (injective) by the non-vanishing of holomorphic differentials at general points.

For a vector bundle $\mathcal{E}$ on $X$, the Atiyah class $\mathrm{At}(\mathcal{E}) \in H^1(X, \mathcal{E}nd(\mathcal{E}) \otimes \Omega^1_X)$ is the obstruction to the existence of an algebraic connection on $\mathcal{E}$. The KS map for any family can be factored through the Atiyah class:

For a deformation $\mathcal{E}_t$ with tangent vector $v \in T_b B$, $\mathrm{KS}(v) = \mathrm{At}(\mathcal{E}) \cup v^*$, where $v^*$ is the pullback of the tangent vector via the classifying map.

For a family of curves $f: \mathcal{C} \to B$ of genus $g$, the period map sends $b$ to the Jacobian $J(C_b)$ with its polarization. The differential is: $d\mathcal{P}: T_b B \xrightarrow{\mathrm{KS}} H^1(C_b, T_{C_b}) \cong H^0(\omega_{C_b}^2)^{\vee} \xrightarrow{\cup} \mathrm{Sym}^2 H^0(\omega_{C_b})^{\vee}$ The composed map is the Petri map (dual). Infinitesimal Torelli holds for non-hyperelliptic curves of genus $g \geq 3$: the period map is an immersion.

For a family of K3 surfaces, the KS map gives $T_b B \xrightarrow{\mathrm{KS}} H^1(S_b, T_{S_b}) \cong H^1(S_b, \Omega^1_{S_b})$ and the period map records the position of $H^{2,0}(S_b) = \mathbb{C}\omega$ in $H^2(S_b, \mathbb{C})$. The local Torelli theorem for K3 surfaces states that the period map is a local isomorphism: the KS map is an isomorphism from $T_b B$ (20-dimensional) to the tangent space of the period domain.

For a smooth curve $C$ of genus $g \geq 2$, $H^2(C, T_C) = 0$, so $\mathrm{ob}_2(\xi_1) = 0$ for all $\xi_1$. Every first-order deformation extends to all orders.

Let $S$ be a smooth surface with $H^2(S, T_S) \neq 0$. A first-order deformation $\xi_1 \in H^1(S, T_S)$ extends to second order if and only if $[\xi_1, \xi_1] = 0$ in $H^2(S, T_S)$. This defines a quadratic cone in $H^1(S, T_S)$, the Kuranishi space being locally the zero set of this quadratic map (plus higher-order terms).

For a Calabi-Yau threefold $X$ (with $\omega_X \cong \mathcal{O}_X$, $H^1(\mathcal{O}_X) = 0$): $H^1(X, T_X) \cong H^1(X, \Omega^2_X) = H^{1,2}(X)$ $H^2(X, T_X) \cong H^2(X, \Omega^2_X) = H^{2,2}(X) \neq 0$ Despite the nonvanishing of the obstruction space, the Bogomolov-Tian-Todorov theorem states that all obstructions vanish: every first-order deformation extends to all orders. The proof uses the $\partial\bar{\partial}$-lemma and the triviality of the canonical bundle to show that $[\xi, \xi]$ is always exact.

A family of genus-$g$ curves $\pi: \mathcal{C} \to S$ determines a map $\phi: S \to \mathcal{M}_g$. The KS map $d\phi_s: T_s S \to H^1(C_s, T_{C_s})$ is computed by the connecting homomorphism of the relative tangent sequence $0 \to T_{\mathcal{C}/S} \to T_{\mathcal{C}} \to \pi^*T_S \to 0$ restricted to the fiber $C_s$. At the stack level, the full tangent complex of $\mathcal{M}_g$ at $[C]$ is concentrated in degree 0 (since $H^0(T_C) = 0$ and $H^2(T_C) = 0$), confirming $\mathcal{M}_g$ is a smooth DM stack.

A Schiffer variation at a point $p \in C$ of a curve $C$ is the first-order deformation corresponding to the KS class supported at $p$. Explicitly, if $z$ is a local coordinate at $p$, the Schiffer variation is the Cech 1-cocycle $\partial/\partial z \in H^1(C, T_C)$ defined on the cover $\lbrace C \setminus \lbrace p \rbrace, U_p \rbrace$. The collection of all Schiffer variations as $p$ varies spans $H^1(C, T_C)$ for a non-hyperelliptic curve.