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 $$\Lambda^{p,p}(M)\cap \Lambda^{2p}(M,{\mathbb R}).$$ A real (1,1)-form $$\omega$$ is called semi-positive (sometimes just positive ), respectively, positive (or positive definite ) if any of the following equivalent conditions holds:


 * 1) $$-\omega$$ is the imaginary part of a positive semidefinite (respectively, positive definite) Hermitian form.
 * 2) For some basis $$dz_1, ... dz_n$$ in the space $$\Lambda^{1,0}M$$ of (1,0)-forms, $$\omega$$ can be written diagonally, as $$\omega = \sqrt{-1} \sum_i \alpha_i dz_i\wedge d\bar z_i,$$ with $$\alpha_i$$ real and non-negative (respectively, positive).
 * 3) For any (1,0)-tangent vector $$v\in T^{1,0}M$$, $$-\sqrt{-1}\omega(v, \bar v) \geq 0$$ (respectively, $$>0$$).
 * 4) For any real tangent vector $$v\in TM$$, $$\omega(v, I(v)) \geq 0$$ (respectively, $$>0$$), where $$I:\; TM\mapsto TM$$ is the complex structure operator.

Positive line bundles
In algebraic geometry, positive definite (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,


 * $$ \bar\partial:\; L\mapsto L\otimes \Lambda^{0,1}(M)$$

its complex structure operator. Then L is equipped with a unique connection preserving the Hermitian structure and satisfying


 * $$\nabla^{0,1}=\bar\partial$$.

This connection is called the Chern connection.

The curvature $$\Theta$$ of the Chern connection is always a purely imaginary (1,1)-form. A line bundle L is called positive if $$\sqrt{-1}\Theta$$ is a positive (1,1)-form. (Note that the de Rham cohomology class of $$\sqrt{-1}\Theta$$ is $$2\pi$$ 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 $$\sqrt{-1}\Theta$$ positive.

Positivity for (p, p)-forms
Semi-positive (1,1)-forms on M form a convex cone. When M is a compact complex surface, $$dim_{\mathbb C}M=2$$, this cone is self-dual, with respect to the Poincaré pairing :$$ \eta, \zeta \mapsto \int_M \eta\wedge\zeta$$

For (p, p)-forms, where $$2\leq p \leq dim_{\mathbb C}M-2$$, there are two different notions of positivity. A form is called strongly positive if it is a linear combination of products of semi-positive forms, with positive real coefficients. A real (p, p)-form $$\eta$$ 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 $$\int_M \eta\wedge\zeta\geq 0 $$.

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