Positive form


In complex geometry, the term positive form refers to several classes of real differential forms of Hodge type .

(1,1)-forms

Real -forms on a complex manifold M are forms which are of type and real, that is, lie in the intersection A real -form is called semi-positive, respectively, positive if any of the following equivalent conditions holds:
  1. is the imaginary part of a positive semidefinite Hermitian form.
  2. For some basis in the space of -forms, can be written diagonally, as with real and non-negative.
  3. For any -tangent vector, .
  4. For any real tangent vector, , where is the complex structure operator.

    Positive line bundles

In algebraic geometry, positive definite -forms arise as curvature forms of ample line bundles. Let L be a holomorphic Hermitian line bundle on a complex manifold,
its complex structure operator. Then L is equipped with a unique connection preserving the Hermitian structure and satisfying
This connection is called the Chern connection.
The curvature of the Chern connection is always a
purely imaginary -form. A line bundle L is called positive if is a positive -form. The Kodaira embedding theorem claims that a positive line bundle is ample, and conversely, any ample line bundle admits a Hermitian metric with positive.

Positivity for ''(p, p)''-forms

Semi-positive -forms on M form a convex cone. When M is a compact complex surface,, this cone is self-dual, with respect to the Poincaré pairing :
For '-forms, where, there are two different notions of positivity. A form is called
strongly positive if it is a linear combination of products of semi-positive forms, with positive real coefficients. A real '-form on an n-dimensional complex manifold M is called weakly positive if for all strongly positive -forms ζ with compact support, we have.
Weakly positive and strongly positive forms form convex cones. On compact manifolds these cones are dual with respect to the Poincaré pairing.