[proofplan] We verify the three defining properties of a Kähler form after restriction to the complex submanifold. Naturality of the [exterior derivative](/theorems/1525) gives closedness. Holomorphicity of the inclusion gives complex-linearity of its differential, which preserves the real $(1,1)$ type under pullback. Finally, injectivity of the differential of an embedded submanifold transfers positivity of $\omega$ on $X$ to positivity of $\iota^*\omega$ on $Y$. [/proofplan]

[step:Use naturality of exterior derivative to prove the restricted form is closed] Let $\alpha := \iota^*\omega \in \Omega^2(Y;\mathbb R)$ be the pullback $2$-form on $Y$. Since $\omega$ is Kähler, it is closed, so $d\omega = 0$. Pullback of differential forms commutes with the exterior derivative, hence $d\alpha = d(\iota^*\omega) = \iota^*(d\omega) = \iota^*0 = 0$. Thus $\alpha$ is closed. [/step]

[step:Use holomorphicity of the inclusion to preserve real type $(1,1)$] Since $Y\subset X$ is a complex submanifold, the inclusion $\iota:Y\to X$ is holomorphic. Therefore, for every $y\in Y$, the differential $d\iota_y:T_yY\to T_{\iota(y)}X$ is complex-linear, meaning $d\iota_y\circ J_Y=J_X\circ d\iota_y$. Because $\omega$ is a real $(1,1)$-form on $X$, it is real-valued and satisfies the $J_X$-invariance identity $\omega_{\iota(y)}(J_X a,J_X b)=\omega_{\iota(y)}(a,b)$ for all $y\in Y$ and all $a,b\in T_{\iota(y)}X$. Let $u,v\in T_yY$. By the definition of pullback and complex-linearity of $d\iota_y$, $\alpha_y(J_Yu,J_Yv)=\omega_{\iota(y)}(d\iota_y(J_Yu),d\iota_y(J_Yv))$. Using $d\iota_y(J_Yu)=J_Xd\iota_y(u)$ and $d\iota_y(J_Yv)=J_Xd\iota_y(v)$, this becomes $\alpha_y(J_Yu,J_Yv)=\omega_{\iota(y)}(J_Xd\iota_y(u),J_Xd\iota_y(v))$. The $(1,1)$ identity for $\omega$ gives $\alpha_y(J_Yu,J_Yv)=\omega_{\iota(y)}(d\iota_y(u),d\iota_y(v))=\alpha_y(u,v)$. Thus $\alpha$ has type $(1,1)$ on $Y$. Since $\omega$ is real-valued and pullback preserves real-valuedness, $\alpha$ is a real $(1,1)$-form. [/step]

[step:Transfer positivity from $\omega$ to the pullback form]Let $y\in Y$ and let $v\in T_yY$ be nonzero. Since $Y\subset X$ is an embedded submanifold, the differential $d\iota_y:T_yY\to T_{\iota(y)}X$ is injective. Hence $d\iota_y(v)\ne 0$. Since $\iota$ is holomorphic, $d\iota_y(J_Yv)=J_Xd\iota_y(v)$. By the definition of pullback, $\alpha_y(v,J_Yv)=\omega_{\iota(y)}(d\iota_y(v),d\iota_y(J_Yv))$. Substituting the complex-linearity identity gives $\alpha_y(v,J_Yv)=\omega_{\iota(y)}(d\iota_y(v),J_Xd\iota_y(v))$. Because $\omega$ is positive on $X$ and $d\iota_y(v)\ne 0$, the right-hand side is positive. Therefore $\alpha_y(v,J_Yv)>0$. Thus $\alpha$ is positive on $Y$.[/step]

[guided]We must prove positivity using only tangent vectors to $Y$. Fix a point $y\in Y$ and choose a nonzero vector $v\in T_yY$. The inclusion map $\iota:Y\to X$ has differential $d\iota_y:T_yY\to T_{\iota(y)}X$. Because $Y$ is an embedded submanifold of $X$, this [linear map](/page/Linear%20Map) is injective. Therefore the nonzero vector $v$ remains nonzero after applying the differential: $d\iota_y(v)\ne 0$. The point of using holomorphicity is that it identifies the complex structures before and after inclusion. Since $\iota$ is holomorphic, its differential is complex-linear: $d\iota_y(J_Yv)=J_Xd\iota_y(v)$. Now evaluate the pullback form $\alpha=\iota^*\omega$ on the pair $(v,J_Yv)$. By the definition of pullback, $\alpha_y(v,J_Yv)=\omega_{\iota(y)}(d\iota_y(v),d\iota_y(J_Yv))$. Using complex-linearity of $d\iota_y$, this becomes $\alpha_y(v,J_Yv)=\omega_{\iota(y)}(d\iota_y(v),J_Xd\iota_y(v))$. The vector $d\iota_y(v)$ is nonzero, and $\omega$ is positive as a Kähler form on $X$. Hence $\omega_{\iota(y)}(d\iota_y(v),J_Xd\iota_y(v))>0$. Combining the two displayed identities gives $\alpha_y(v,J_Yv)>0$. Since $y\in Y$ and $0\ne v\in T_yY$ were arbitrary, $\alpha$ is positive on $Y$. If $Y$ has complex dimension $0$, this condition is vacuous because there are no nonzero tangent vectors.[/guided]

[step:Conclude that the pullback is a Kähler form on the submanifold] We have shown that $\alpha=\iota^*\omega$ is closed, real of type $(1,1)$, and positive on $Y$. These are precisely the defining properties of a Kähler form on the complex manifold $Y$. Therefore $\iota^*\omega$ is a Kähler form on $Y$. [/step]