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For the underlying Riemannian metric $g$ and complex structure $J$, the fundamental form is $\omega (u,v)=g(Ju,v)$. It is a real (1, 1)-form. In complex dimension $n$, the compatible orientation has Riemannian volume form ${\omega }^n/n!$, and its Hodge star operator satisfies

Let $e$ be a holomorphic local frame of a Hermitian holomorphic line bundle and put $h=h(e,e)$. The Chern connection and its curvature are locallyThus $F_{\nabla }$ has type $(1,1)$. Because the connection is unitary, ${\bar{F}}_{\nabla }=-F_{\nabla }$, so $i{F}_{\nabla }$ is a real (1, 1)-form.

For a holomorphic line bundle with Hermitian metric $h$, its normalized Chern curvature iswhere $e$ is a holomorphic local frame and $h_e=h(e,e)>0$. A change of frame multiplies $h_e$ by the squared absolute value of a holomorphic unit; its logarithm has zero mixed derivative. The local expressions therefore define a closed real (1, 1)-form representing the First Chern class.

Positive curvature is an additional condition on the metric, distinct from positivity of each fibre norm. On the trivial line over $\mathbb{C}$, the metric $h_e=1$ is positive definite but has zero curvature. The metric $h_e=e^{-|z{|}^2}$ instead gives ${\omega }_{(L,h)}=idz\wedge d\bar{z}/(2\pi )$, a positive real (1, 1)-form. Globally, admitting such a positively curved metric is the definition of a positive holomorphic line bundle.

Replacing $h$ by $e^{-u}h$ adds $(i/(2\pi ))\partial \bar{\partial }u$. Thus the normalized Chern curvature changes by an exact differential form, while its First Chern class stays fixed.

Use the integral normalization of the Chern connection curvature. For a holomorphic local frame $e$, put $h_e=h(e,e)>0$. The fundamental form of a Hermitian holomorphic line bundle isThis follows from the local formula for the Chern connection on a line bundle, $F_h=\bar{\partial }\partial \log h_e$. It is a closed real (1, 1)-form representing the First Chern class. If $h_e=e^{-\phi }$, the formula becomes $(i/(2\pi ))\partial \bar{\partial }\phi$.

The Hermitian metric is positive when this is a positive real (1, 1)-form: for every nonzero $v\in T^{1,0}M$,Equivalently, the matrix $({\partial }^2\phi /\partial z_j\partial {\bar{z}}_k)$ is a positive-definite matrix in every frame. Such a metric makes $\mathcal{L}$ a positive holomorphic line bundle. Omitting the normalization factor $2\pi$ changes none of these positivity conditions.

A real (1, 1)-form is a form $\phi \in {\Omega }^{1,1}(X)$ satisfying $\bar{\phi }=\phi$. In holomorphic coordinates it has the formIt is a positive real (1, 1)-form whenfor every nonzero tangent vector $\xi$ of type $(1,0)$, equivalently when the Hermitian matrix $(h_{j\bar{k}})$ is positive definite.

A holomorphic local trivialization of a holomorphic line bundle $L\to X$ is equivalently a nowhere-zero holomorphic local frame $e$. A connection $A$ is unitary when it preserves the fiberwise Hermitian inner product:The Chern connection is the unique unitary connection whose $(0,1)$ part is the bundle's Dolbeault partial connection ${\bar{\partial }}_L$.

In a holomorphic frame, put $h=h(e,e)$. The local formula for the Chern connection on a line bundle isThe curvature therefore has type $(1,1)$. Since a unitary connection has imaginary curvature, $\overline{F(A)}=-F(A)$, and henceThus $iF(A)$ is a real $(1,1)$-form.

For connections $A$ on $L$ and $\hat{A}$ on $\hat{L}$, the tensor product connection is defined on decomposable local sections byThe curvature of a tensor product connection on line bundles is additive:Consequentlywhich is positive whenever both summands are positive.

For the final assertion, simultaneously diagonalize the positive Hermitian matrices of $\phi$ and $\psi$ by congruence at the chosen point. In the resulting coframe,A direct wedge-product calculation givesIf $\xi$ and $\eta$ are linearly independent, at least one of these $2\times 2$ minors is nonzero. Every coefficient is positive, so the sum is strictly positive. This is wedge positivity for two positive (1, 1)-forms.