[proofplan] The proof is a sheaf-theoretic translation of the elementary linear algebra fact that contraction with a nonzero volume form identifies a [vector space](/page/Vector%20Space) with its space of alternating covectors of codimension one. Since $\Omega$ is holomorphic and nowhere vanishing, this pointwise identification varies holomorphically and gives an isomorphism $T_X \cong \Omega_X^{n-1}$. Taking sheaf cohomology gives $H^1(X,T_X)\cong H^1(X,\Omega_X^{n-1})$, and Hodge theory for compact Kähler manifolds identifies the latter group with Dolbeault cohomology $H^{n-1,1}(X)$. Finally, connectedness and compactness force any other holomorphic volume form to be a nonzero scalar multiple of $\Omega$. [/proofplan] [step:Define the contraction morphism of holomorphic vector bundles] Let $T_X$ denote the holomorphic tangent bundle of $X$, let $K_X=\Omega_X^n$ denote the canonical bundle, and let $\Omega_X^{n-1}$ denote the bundle of holomorphic $(n-1)$-forms. Define the bundle morphism \begin{align*}\Phi_\Omega:T_X &\longrightarrow \Omega_X^{n-1}\end{align*} by setting, for each point $x\in X$, \begin{align*}(\Phi_\Omega)_x:T_{X,x} &\longrightarrow \Lambda^{n-1}T_{X,x}^*\end{align*} to be the [linear map](/page/Linear%20Map) \begin{align*} v &\longmapsto \iota_v\Omega_x, \end{align*} where $\iota_v\Omega_x$ denotes contraction of the alternating $n$-form $\Omega_x\in \Lambda^nT_{X,x}^*$ with the vector $v\in T_{X,x}$. In a holomorphic coordinate chart $(U,z)$ with coordinates $z_1,\dots,z_n$, write \begin{align*}\Omega|_U=f\,dz_1\wedge \cdots \wedge dz_n\end{align*} for a [holomorphic function](/page/Holomorphic%20Function) $f:U\to \mathbb C$. Since $\Omega$ is nowhere vanishing, $f(x)\neq 0$ for every $x\in U$. For a local holomorphic vector field \begin{align*} V=\sum_{j=1}^n V_j\,\partial_{z_j}, \end{align*} where each $V_j:U\to \mathbb C$ is holomorphic, contraction gives \begin{align*}\Phi_\Omega(V)=\sum_{j=1}^n (-1)^{j-1} fV_j\,dz_1\wedge \cdots \wedge dz_{j-1}\wedge dz_{j+1}\wedge \cdots \wedge dz_n.\end{align*} Thus the coefficients of $\Phi_\Omega(V)$ are holomorphic whenever the coefficients of $V$ are holomorphic, so $\Phi_\Omega$ is a holomorphic vector bundle morphism. [/step] [step:Prove that the contraction morphism is pointwise an isomorphism] Fix a point $x\in X$, and set $V=T_{X,x}$. Since $\Omega$ is nowhere vanishing, $\Omega_x$ is a nonzero element of $\Lambda^nV^*$. Choose a basis $e_1,\dots,e_n$ of $V$, and let $\theta_1,\dots,\theta_n$ be its [dual basis](/theorems/414). There is a scalar $a\in\mathbb C^\times$ such that \begin{align*} \Omega_x=a\,\theta_1\wedge\cdots\wedge\theta_n. \end{align*} For each index $j\in\{1,\dots,n\}$, contraction gives \begin{align*}\iota_{e_j}\Omega_x=(-1)^{j-1}a\,\theta_1\wedge\cdots\wedge\theta_{j-1}\wedge\theta_{j+1}\wedge\cdots\wedge\theta_n.\end{align*} The forms on the right, as $j$ varies from $1$ to $n$, form a basis of $\Lambda^{n-1}V^*$ because they are nonzero scalar multiples of the standard basis of $(n-1)$-fold wedge products obtained by omitting one dual basis vector. Therefore $(\Phi_\Omega)_x$ sends a basis of $V$ to a basis of $\Lambda^{n-1}V^*$, so $(\Phi_\Omega)_x$ is a linear isomorphism. Since this holds for every $x\in X$, the holomorphic vector bundle morphism $\Phi_\Omega:T_X\to \Omega_X^{n-1}$ is an isomorphism of holomorphic vector bundles. [guided] The key point is purely linear algebra. At a fixed point $x\in X$, the fibre $T_{X,x}$ is an $n$-dimensional complex vector space, and the fibre of $\Omega_X^{n-1}$ is $\Lambda^{n-1}T_{X,x}^*$. A nonzero element of $\Lambda^nT_{X,x}^*$ is a volume form on this vector space, and contraction with such a volume form should convert one vector into the complementary alternating covector. Let $V=T_{X,x}$. Choose a basis $e_1,\dots,e_n$ of $V$, and let $\theta_1,\dots,\theta_n$ be the dual basis of $V^*$. Since $\Lambda^nV^*$ is one-dimensional and $\Omega_x\neq 0$, there is a nonzero scalar $a\in\mathbb C^\times$ satisfying \begin{align*} \Omega_x=a\,\theta_1\wedge\cdots\wedge\theta_n. \end{align*} Now compute the contraction on each basis vector. By the definition of interior product, \begin{align*} \iota_{e_j}\Omega_x=(-1)^{j-1}a\,\theta_1\wedge\cdots\wedge\theta_{j-1}\wedge\theta_{j+1}\wedge\cdots\wedge\theta_n. \end{align*} The sign records the number of one-forms that $e_j$ must pass before pairing with $\theta_j$; the scalar $a$ is nonzero because $\Omega_x$ is nonzero. The target space $\Lambda^{n-1}V^*$ has dimension $n$, with basis given by the $(n-1)$-forms obtained from $\theta_1\wedge\cdots\wedge\theta_n$ by omitting one $\theta_j$. The displayed formula shows that $(\Phi_\Omega)_x$ sends the basis $e_1,\dots,e_n$ of $V$ to nonzero scalar multiples of this basis of $\Lambda^{n-1}V^*$. Hence $(\Phi_\Omega)_x$ is a linear isomorphism. Because the same argument applies at every point $x\in X$, and because the preceding step already proved that $\Phi_\Omega$ is holomorphic, $\Phi_\Omega:T_X\to\Omega_X^{n-1}$ is an isomorphism of holomorphic vector bundles. [/guided] [/step] [step:Pass the bundle isomorphism to first sheaf cohomology] An isomorphism of holomorphic vector bundles induces an isomorphism of the corresponding sheaves of holomorphic sections. Therefore $\Phi_\Omega:T_X\to\Omega_X^{n-1}$ induces an isomorphism on sheaf cohomology: \begin{align*} H^1(\Phi_\Omega):H^1(X,T_X)\longrightarrow H^1(X,\Omega_X^{n-1}). \end{align*} Since $\Phi_\Omega$ is an isomorphism of sheaves, the induced map $H^1(\Phi_\Omega)$ is an isomorphism of complex vector spaces. [/step] [step:Identify the sheaf cohomology group with Dolbeault cohomology] Let $\mathcal O_X$ denote the holomorphic line bundle whose local holomorphic sections are holomorphic functions on $X$, equipped with the constant Hermitian metric in local frames. Apply [citetheorem:9104] with $E=\mathcal O_X$, $p=n-1$, and $q=1$. The hypotheses are satisfied because $X$ is compact Kähler and $\mathcal O_X$ is a Hermitian holomorphic vector bundle. The theorem identifies harmonic $\mathcal O_X$-valued $(n-1,1)$-forms, hence ordinary harmonic $(n-1,1)$-forms, with the sheaf cohomology group $H^1(X,\Omega_X^{n-1})$. We denote this Dolbeault cohomology group, equivalently the corresponding Hodge summand, by \begin{align*} H^{n-1,1}(X). \end{align*} Thus there is a canonical isomorphism \begin{align*} H^1(X,\Omega_X^{n-1})\cong H^{n-1,1}(X). \end{align*} Composing this isomorphism with $H^1(\Phi_\Omega)$ gives \begin{align*} H^1(X,T_X)\cong H^{n-1,1}(X). \end{align*} [/step] [step:Show that changing the holomorphic volume form only rescales the isomorphism] Let $\Omega'\in H^0(X,K_X)$ be another nowhere-vanishing holomorphic volume form. Since $\Omega$ has no zeros, there is a unique holomorphic function $r:X\to\mathbb C^\times$ such that $\Omega'=r\Omega$. Because $X$ is compact, the holomorphic function $r:X\to\mathbb C$ attains a maximum for $|r|$; by the [maximum modulus principle](/theorems/491) on the connected compact complex manifold $X$, $r$ is constant. Thus there is a unique scalar $c\in\mathbb C^\times$ such that $\Omega'=c\Omega$. For every point $x\in X$ and every vector $v\in T_{X,x}$, \begin{align*}\iota_v\Omega'_x=\iota_v(c\Omega_x)=c\,\iota_v\Omega_x.\end{align*} Hence $\Phi_{\Omega'}=c\,\Phi_\Omega$, and the induced map on $H^1(X,T_X)$ is multiplied by the same nonzero scalar $c$. Therefore the displayed isomorphism depends on the chosen volume form by nonzero scalar multiplication, while the resulting equality of dimensions is intrinsic. This proves the theorem. [/step]