Positive form

(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 : Λ p , p ( M ) ∩ Λ 2 p ( M , R ) . A real (1,1)-form ω is called positive if any of the following equivalent conditions hold

1. − ω is an imaginary part of a positive (not necessarily positive definite) Hermitian form.
2. For some basis d z 1 , . . . d z n in the space Λ 1 , 0 M of (1,0)-forms, − 1 ω can be written diagonally, as − 1 ω = ∑ i α i d z i ∧ d z ¯ i , with α i real and non-negative.
3. For any (1,0)-tangent vector v ∈ T 1 , 0 M , − − 1 ω ( v , v ¯ ) ≥ 0
4. For any real tangent vector v ∈ T M , ω ( v , I ( v ) ) ≥ 0 , where I : T M ↦ T M is the complex structure operator.

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 Θ 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 − 1 Θ positive.

Positivity for (p, p)-forms

Positive (1,1)-forms on M form a convex cone. When M is a compact complex surface, d i m C M = 2 , this cone is self-dual, with respect to the Poincaré pairing : η , ζ ↦ ∫ M η ∧ ζ

For (p, p)-forms, where 2 ≤ p ≤ d i m C M − 2 , 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 η 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 ∫ M η ∧ ζ ≥ 0 .

Weakly positive and strongly positive forms form convex cones. On compact manifolds these cones are dual with respect to the Poincaré pairing.