Positive form

In complex geometry, the term "positive form" refers to several classes of real differential forms of Hodge type "(p, p)".
(1,1)-forms.
Real ("p","p")-forms on a complex manifold "M" are forms which are of type ("p","p") and real, that is, lie in the intersection :formula_1 A real (1,1)-form formula_2 is called positive if any of the following equivalent conditions hold
Positive line bundles.
In algebraic geometry, positive (1,1)-forms arise as curvature forms of ample line bundles (also known as "positive 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 formula_16 of a Chern connection is always a
purely imaginary (1,1)-form. A line bundle "L" is called "positive" if
is a positive definite (1,1)-form. The Kodaira embedding theorem claims that a positive line bundle is ample, and conversely, any ample line bundle admits a Hermitian metric with formula_17 positive.
Positivity for "(p, p)"-forms.
Positive (1,1)-forms on "M" form a convex cone. When "M" is a compact complex surface, formula_19, this cone is self-dual, with respect to the Poincaré pairing :formula_20
For "(p, p)"-forms, where formula_21, there are two different notions of positivity. A form is called
strongly positive if it is a linear combination of products of positive forms, with positive real coefficients. A real "(p, p)"-form formula_22 on an "n"-dimensional complex manifold "M" is called weakly positive if for all strongly positive "(n-p, n-p)"-forms ζ with compact support, we have formula_23.
Weakly positive and strongly positive forms form convex cones. On compact manifolds these cones are dual with respect to the Poincaré pairing.