Positive form
# 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:

* $-\omega$ is the imaginary part of a positive semidefinite (respectively, positive definite) Hermitian form.
* For some basis $dz_{1},...dz_{n}$ in the space $\Lambda ^{1,0}M$ of (1,0)-forms, ${\sqrt {-1}}\omega$ can be written diagonally, as ${\sqrt {-1}}\omega =\sum _{i}\alpha _{i}dz_{i}\wedge d{\bar {z}}_{i},$ with $\alpha _{i}$ real and non-negative (respectively, positive).
* For any (1,0)-tangent vector $v\in T^{1,0}M$, $-{\sqrt {-1}}\omega (v,{\bar {v}})\geq 0$ (respectively, $>0$).
* 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.

