[proofplan] We use differentiable local triviality of a proper submersion to identify nearby fibres with the fixed smooth manifold underlying the central fibre. Starting from a Kähler form $\omega_0$ on the central fibre, the Kodaira-Spencer analytic stability argument constructs closed real $(1,1)$-forms for the nearby complex structures that depend smoothly on the parameter and converge to $\omega_0$ in the smooth topology. Since positivity of a real $(1,1)$-form is an open condition in the smooth topology, these nearby forms are Kähler forms. Pushing them through the fibre identifications gives Kähler forms on the nearby holomorphic fibres. [/proofplan]

[step:Trivialize the smooth family near the central fibre] Regard $X:=X_0=\pi^{-1}(0)$ as the smooth manifold underlying the central fibre, and let $J_0:TX\to TX$ be its complex structure. Choose a Kähler form $\omega_0$ on $(X,J_0)$, so $\omega_0$ is a closed real smooth $2$-form of type $(1,1)$ and $\omega_0(v,J_0v)>0$ for every nonzero tangent vector $v\in TX$. Write $A^{1,1}(X,J_0;\mathbb R)$ for the real smooth $(1,1)$-forms on $(X,J_0)$ and $A^2(X;\mathbb R)$ for the real smooth $2$-forms on $X$. By Ehresmann's proper submersion theorem, applied to the proper smooth submersion underlying $\pi$, there is an open neighbourhood $B_1\subset B$ of $0$ and a smooth diffeomorphism $\Phi:X\times B_1\to \pi^{-1}(B_1)$ satisfying $\pi(\Phi(x,b))=b$ for all $(x,b)\in X\times B_1$ and $\Phi(x,0)=x$ after identifying $X$ with $\mathcal X_0$. For each $b\in B_1$, define the smooth fibre-identification map $\Phi_b:X\to \mathcal X_b$ by $\Phi_b(x)=\Phi(x,b)$. Pulling the complex structure of $\mathcal X_b$ back by $\Phi_b$ gives a smooth family of integrable almost-complex structures $J_b:TX\to TX$, with $J_0$ equal to the original complex structure on $X$. The pulled-back form $\omega_0$ is a fixed closed real $2$-form on the smooth manifold $X$, so in particular $\omega_0\in A^2(X;\mathbb R)$. To prove that $\mathcal X_b$ is Kähler, it is enough to construct, for all $b$ sufficiently close to $0$, a closed real $2$-form $\omega_b$ on $X$ that has type $(1,1)$ with respect to $J_b$ and is positive with respect to $J_b$, because then $(\Phi_b^{-1})^*\omega_b$ is a Kähler form on $\mathcal X_b$. [/step]

[step:Apply the Kodaira-Spencer analytic stability lemma to obtain nearby closed $(1,1)$ forms]We quote the Kodaira-Spencer analytic stability lemma for Kähler forms as the external analytic input. It states that if $(J_b)_{b\in B_1}$ is a smooth family of integrable complex structures on the compact smooth manifold $X$ and $\omega_0$ is a Kähler form for $J_0$, then, after shrinking $B_1$ to an open neighbourhood $B_2$ of $0$, there is a smooth family of real $2$-forms $b\mapsto \omega_b$ on $X$ such that $\omega_b$ is closed and of type $(1,1)$ with respect to $J_b$ for every $b\in B_2$, and $\omega_b\to\omega_0$ in the $C^\infty$ topology as $b\to0$. This lemma is the elliptic $\partial\bar\partial$-correction step in the deformation theory of compact Kähler manifolds; it does not assert that the fixed de Rham class $[\omega_0]$ remains of type $(1,1)$ under deformation. The hypotheses hold because $X$ is compact, the structures $J_b$ are integrable and depend smoothly on $b$, and $\omega_0$ is a Kähler form for $J_0$. Hence there is an open neighbourhood $B_2\subset B_1$ of $0$ and a smooth family of real $2$-forms $b\mapsto \omega_b$ on $X$ such that, for every $b\in B_2$, $d\omega_b=0$, $\omega_b$ has type $(1,1)$ with respect to $J_b$, and $\omega_b\to\omega_0$ in the $C^\infty$ topology as $b\to0$.[/step]

[guided]The transported form $\omega_0$ remains closed on the underlying smooth manifold $X$, but it need not remain of type $(1,1)$ for the changed complex structure $J_b$. We therefore do not assert that the fixed de Rham class $[\omega_0]$ remains a $(1,1)$ class; individual Hodge classes can leave $H^{1,1}$ under deformation. The Kodaira-Spencer analytic stability lemma supplies the correct replacement, and in this proof it is used as an external analytic theorem. Its input is a compact smooth manifold $X$, a smooth family of integrable complex structures $J_b$, and a Kähler form $\omega_0$ for the central complex structure $J_0$. These hypotheses are satisfied here: compactness comes from the central fibre, the family $J_b$ is obtained by pulling back the holomorphic fibre complex structures through the smooth trivialisation, and $\omega_0$ was chosen to be Kähler on $(X,J_0)$. The conclusion is that, after replacing $B_1$ by a smaller open neighbourhood $B_2$ of $0$, there are real smooth $2$-forms $\omega_b$ on $X$ with $d\omega_b=0$ and with $\omega_b$ of type $(1,1)$ for $J_b$. Moreover the family is smooth in $b$ and satisfies $\omega_b\to\omega_0$ in the $C^\infty$ topology as $b\to0$. Conceptually, the analytic argument corrects the transported central form by solving an elliptic $\partial\bar\partial$-type equation depending smoothly on the parameter. The $\partial\bar\partial$-lemma on the central Kähler fibre gives the solvability at $b=0$, and elliptic estimates preserve solvability for all sufficiently small $b$.[/guided]

[step:Shrink the base so the corrected forms remain positive] It remains to show that $\omega_b$ is positive for all $b$ close to $0$. Choose any Riemannian metric $g$ on the compact smooth manifold $X$, and let $S_gX\subset TX$ denote its unit sphere bundle. Define the [continuous function](/page/Continuous%20Function) $F_0:S_gX\to \mathbb R$ by $F_0(v)=\omega_0(v,J_0v)$. Since $\omega_0$ is Kähler, $F_0(v)>0$ for every $v\in S_gX$. Since $S_gX$ is compact, the minimum $m:=\min_{v\in S_gX}F_0(v)$ exists and satisfies $m>0$. The family $F:S_gX\times B_2\to \mathbb R$, $(v,b)\mapsto \omega_b(v,J_bv)$, is continuous and satisfies $F(v,0)=F_0(v)$. By compactness of $S_gX$ and continuity of $F$, after shrinking $B_2$ to an open neighbourhood $U\subset B_2$ of $0$, we have $F(v,b)>\frac{m}{2}$ for every $v\in S_gX$ and every $b\in U$. Therefore, for every $b\in U$ and every nonzero vector $w\in TX$, writing $v=w/|w|_g$, we obtain \begin{align*} \omega_b(w,J_bw)=|w|_g^2\omega_b(v,J_bv)>0. \end{align*} Thus $\omega_b$ is positive as a real $(1,1)$-form on $(X,J_b)$ for every $b\in U$. [/step]

[step:Push the Kähler forms back to the holomorphic fibres] Fix $b\in U$. The form $\omega_b$ is closed, real, of type $(1,1)$ with respect to $J_b$, and positive with respect to $J_b$. Define a real $2$-form on the fibre $\mathcal X_b$ by $\widetilde{\omega}_b := (\Phi_b^{-1})^*\omega_b$. Since pullback by the diffeomorphism $\Phi_b^{-1}:\mathcal X_b\to X$ commutes with exterior differentiation, $d\widetilde{\omega}_b=0$. Since $J_b$ is the pullback of the complex structure on $\mathcal X_b$, the form $\widetilde{\omega}_b$ has type $(1,1)$ on $\mathcal X_b$. Positivity is also preserved by the same identification, so $\widetilde{\omega}_b$ is a Kähler form on $\mathcal X_b$. Hence every fibre $\mathcal X_b$ with $b\in U$ is Kähler, which proves the theorem. [/step]