Hermitian metrics and complex differential forms

In holomorphic coordinates write the positive Hermitian coefficient matrix as G, with entries indexed by one holomorphic and one antiholomorphic index. Define the real metric by twice the real part of its Hermitian tensor, and define the fundamental form by applying J to the first argument of that real metric. The resulting form has the coordinate expression below. With these choices, its evaluation on a nonzero real vector and its J-rotation is positive. Both the real metric and the form determine one another once J is fixed.

For one complex coordinate z equal to x plus iy, the wedge of dz with its conjugate is minus twice i times the oriented real area form. Thus coefficient one half gives the usual Euclidean metric and usual oriented area form. Coefficient one instead doubles both. Neither convention is intrinsically better, but mixing them would change Laplacian factors and curvature comparisons. We use the displayed convention throughout the module and state any rescaling explicitly when comparing examples or sources.

Complex differential forms split into bidegrees: a type p-q form has p holomorphic and q antiholomorphic differentials. Conjugation exchanges the two indices. A real two-form of type one-one can therefore contain factors of i while still being real, because conjugation reverses the wedge order as well. Wedge products add the bidegrees and obey the usual total-degree sign. The fundamental form is real and of type one-one, while a top holomorphic form has type n-zero and belongs to the canonical bundle.

On an integrable complex manifold, exterior differentiation splits as partial plus bar-partial, increasing the respective type indices. This simple two-piece decomposition relies on integrability; general almost complex structures can introduce additional type components. For the Hermitian metric, the top wedge of the fundamental form divided by factorial is the positive Riemannian volume form. A determinant calculation in a coordinate chart gives its density. It is the full determinant, rather than only diagonal entries, that accounts for correlations between coordinate directions.

A smooth positive Hermitian matrix field always defines a Hermitian metric. It becomes Kähler only when its fundamental form is closed. In complex dimension one, a two-form is already of top real degree, so closedness is automatic. In higher dimensions, derivatives of the coefficients impose nontrivial conditions. For example, letting the second diagonal coefficient depend on the first coordinate produces a positive metric whose derivative generally has a nonzero three-form component. Pointwise positivity cannot detect this variation.

At a single tangent space, an invertible real coordinate matrix has a positive Gram matrix given by transpose times itself. Its singular values determine the Gram eigenvalues, so a matrix diagram can visualize positive quadratic forms and anisotropy. This does not create a globally defined Hermitian metric field or check its closedness. The useful lesson is narrower: positive definiteness is a pointwise statement, while the Kähler condition involves spatial derivatives that cannot be inferred from one selected matrix's determinant.