In complex geometry, the term positive form refers to several classes of real differential forms of Hodge type (p, p).
Let L be a holomorphic Hermitian line bundle on a complex manifold, its complex structure operator.
of the Chern connection is always a purely imaginary (1,1)-form.
A line bundle L is called positive if
(Note that the de Rham cohomology class of
times the first Chern class of L.) The Kodaira embedding theorem claims that a positive line bundle is ample, and conversely, any ample line bundle admits a Hermitian metric with
Semi-positive (1,1)-forms on M form a convex cone.
, this cone is self-dual, with respect to the Poincaré pairing :
[5] A form is called strongly positive if it is a linear combination of products of semi-positive forms, with positive real coefficients.
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
Weakly positive and strongly positive forms form convex cones.
On compact manifolds these cones are dual with respect to the Poincaré pairing.