Kähler forms and Kähler identities

A Kähler form is a real closed positive one-one form on a complex manifold. Its positivity means evaluation on a nonzero real vector and its J-rotation is positive. Locally it can be written as i times partial bar-partial of a real potential K. With our normalization, the Hermitian coefficient matrix is the complex Hessian of K, and the Euclidean real metric uses K equal to one half the squared complex norm. Strict plurisubharmonicity of K is exactly positivity of that Hessian.

A potential is local: changing it by the real part of a holomorphic function leaves the form unchanged. On a positive-dimensional compact Kähler manifold a global smooth strictly plurisubharmonic potential for the entire Kähler form cannot exist. Such a potential would make the form globally exact, contradicting its positive top-volume integral by Stokes. This does not forbid a global potential for the difference between two cohomologous Kähler forms, which is a central consequence of the compact Kähler partial-bar-partial lemma.

Let L wedge a form with omega and let Lambda be its metric adjoint. Let partial-star and bar-partial-star denote the adjoints of the two differentials, and define each Laplacian as differential times adjoint plus adjoint times differential. This convention makes the Laplacians nonnegative in the global inner product on a compact boundaryless manifold. In a Euclidean chart the real Laplacian on functions is therefore minus the sum of second real derivatives, rather than the opposite analyst's sign.

The displayed commutators fix the factors of i and imply equality of the two complex Laplacians, each one half the real Laplacian. The Kähler condition removes extra torsion terms that occur for a general Hermitian metric. These are identities of differential operators; a local calculation can verify them in Kähler normal coordinates, but we invoke that full calculation rather than pretending one example proves it. Compactness is not required for the local identities, although global spectral and Hodge conclusions require additional analytic hypotheses.

A form is primitive when Lambda annihilates it. This is an algebraic condition at each tangent space and does not mean the form itself vanishes. In complex dimension two, the difference of the area forms in two orthonormal complex directions is a nonzero primitive one-one form. Wedge multiplication by the total Kähler form annihilates that particular middle-degree form. Lefschetz algebra organizes such decompositions, and the Kähler identities make these algebraic operators interact with harmonic representatives.

Equality of Laplacians implies that a real-Laplacian harmonic form has harmonic components of every bidegree on a compact Kähler manifold. The next lesson turns this into a decomposition of de Rham cohomology. To see the numerical factor without invoking topology, one may inspect a single flat Fourier mode: its real heat decay and complex-Laplacian heat decay differ by a factor of two in rate. The one-dimensional heat instrument illustrates those rates for its sine mode, while the full geometric operators remain calculations on the chosen manifold.