A Hermitian manifold is a complex manifold whose tangent bundle carries a smoothly varying positive-definite Hermitian form. Its real part is a Riemannian metric invariant under the complex structure .
Exterior multiplication by the fundamental -form of a Hermitian manifold raises bidegree by . Its pointwise formal adjoint lowers bidegree by . On a Kähler manifold this is the Lefschetz operator of a Kähler manifold, but its pointwise algebraic properties do not require the form to be closed.
The Hermitian Lefschetz operator has injective powers in the indicated range. The commutator formula for powers of the Lefschetz operator proves this by induction on : a form in the kernel can be written as using injectivity of the smaller power, and the lower-degree form is then in the kernel of another power covered by the induction hypothesis.
For the underlying Riemannian metric and complex structure , the fundamental form is . It is a real (1, 1)-form. In complex dimension , the compatible orientation has Riemannian volume form , and its Hodge star operator satisfies
In an adapted real orthonormal coframe, the fundamental form of a Hermitian manifold is and the Riemannian volume form is . The complex-linear Hodge star sends each summand to the product of all the other oriented two-dimensional factors. Summation gives the displayed formula, valid for every Hermitian manifold, without a Kähler assumption.
A differential form of type (p, q) of type is real when . Locally this is equivalent tofor a Hermitian matrix .
A real -form is positive whenfor every nonzero tangent vector of type . Equivalently, the local Hermitian matrix in the representation is positive definite.
If and are positive real -forms and are linearly independent tangent vectors of type , thenAfter simultaneous diagonalization by congruence, with positive diagonal entries , the left side iswhich is positive because some minor is nonzero.
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A Hermitian manifold is a type of complex manifold equipped with a Riemannian metric that is compatible with the complex structure. More formally, a Hermitian manifold consists of the following components: 1. **Complex Manifold**: A manifold \( M \) that is equipped with an atlas of charts where the transition functions are holomorphic mappings. This means that the local coordinates can be expressed in terms of complex variables.