In complex geometry

Complex geometry

In mathematics, complex geometry is the study of complex manifolds and functions of many complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric chapters of complex analysis....

, 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
A real (1,1)-form
is called positive if any of the
following equivalent conditions hold

1. is an imaginary part of a positive (not necessarily positive definite) Hermitian form.
2. For some basis in the space of (1,0)-forms, can be written diagonally, as with real and non-negative.
3. For any (1,0)-tangent vector ,
4. For any real tangent vector , , where is the complex structure operator.

Positive line bundles

In algebraic geometry, positive (1,1)-forms arise as curvature
forms of ample line bundle

Ample line bundle

In algebraic geometry, a very ample line bundle is one with enough global sections to set up an embedding of its base variety or manifold M into projective space. An ample line bundle is one such that some positive power is very ample...

s (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

Hermitian connection

In mathematics, the Hermitian connection \nabla, also called the Chern connection, is the unique connection on a Hermitian manifold that satisfies the following conditions,# It preserves the metric g, i.e., \nabla g=0....

.

The curvature 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

Kodaira embedding theorem

In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds...

 claims that a positive line bundle is ample, and conversely, any ample line bundle

Ample line bundle

In algebraic geometry, a very ample line bundle is one with enough global sections to set up an embedding of its base variety or manifold M into projective space. An ample line bundle is one such that some positive power is very ample...

 admits a Hermitian metric with positive.

Positivity for (p, p)-forms

Positive (1,1)-forms on M form a convex cone

Convex cone

In linear algebra, a convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.-Definition:...

.
When M is a compact complex surface,
, this cone is
self-dual, with respect
to the Poincaré pairing

For (p, p)-forms, where ,
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 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.